Lesson 6 of 24
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Exponential and Logarithmic Functions

Exponential Functions

Exponential functions arise in many applications. One common example is population growth.

For example, if a population starts with P_0 individuals and then grows at an annual rate of 2\%, its population after 1 year isP(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02).

Its population after 2 years isP(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2.

In general, its population after t years isP(t)=P_0(1.02)^t,

which is an exponential function. More generally, any function of the form f(x)=b^x, where b>0, \, b \ne 1″ height=”17″ width=”90″>, is an exponential function with <strong>base</strong> <img loading= and exponent x. Exponential functions have constant bases and variable exponents. Note that a function of the form f(x)=x^b for some constant b is not an exponential function but a power function.

To see the difference between an exponential function and a power function, we compare the functions y=x^2 and y=2^x. In (Figure), we see that both 2^x and x^2 approach infinity as x \to \infty. Eventually, however, 2^x becomes larger than x^2 and grows more rapidly as x \to \infty. In the opposite direction, as x \to −\infty, \, x^2 \to \infty, whereas 2^x \to 0. The line y=0 is a horizontal asymptote for y=2^x.

\mathbf{x}-3-2-10123456
\mathbf{x^2}9410149162536
\mathbf{2^x}1/81/41/21248163264

In (Figure), we graph both y=x^2 and y=2^x to show how the graphs differ.

An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is “y = x squared”, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is “y = 2 to the power of x”, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.
Figure 1. Both 2^x and x^2 approach infinity as x \to \infty, but 2^x grows more rapidly than x^2. As x \to −\infty, \, x^2 \to \infty, whereas 2^x \to 0.

Evaluating Exponential Functions

Recall the properties of exponents: If x is a positive integer, then we define b^x=b·b \cdots b (with x factors of b). If x is a negative integer, then x=−y for some positive integer y, and we define b^x=b^{−y}=1/b^y. Also, b^0 is defined to be 1. If x is a rational number, then x=p/q, where p and q are integers and b^x=b^{p/q}=\sqrt[q]{b^p}. For example, 9^{3/2}=\sqrt{9^3}=27. However, how is b^x defined if x is an irrational number? For example, what do we mean by 2^{\sqrt{2}}? This is too complex a question for us to answer fully right now; however, we can make an approximation. In (Figure), we list some rational numbers approaching \sqrt{2}, and the values of 2^x for each rational number x are presented as well. We claim that if we choose rational numbers x getting closer and closer to \sqrt{2}, the values of 2^x get closer and closer to some number L. We define that number L to be 2^{\sqrt{2}}.

\mathbf{x}1.41.411.4141.41421.414211.414213
\mathbf{2^x}2.6392.657372.664752.6651192.6651382.665143

Bacterial Growth

Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is n(4)=1000·2. The number of bacteria after 8 hours is n(8)=n(4)·2=1000·2^2. In general, the number of bacteria after 4m hours is n(4m)=1000·2^m. Letting t=4m, we see that the number of bacteria after t hours is n(t)=1000·2^{t/4}. Find the number of bacteria after 6 hours, 10 hours, and 24 hours.

Solution

The number of bacteria after 6 hours is given by n(6)=1000·2^{6/4} \approx 2828 bacteria. The number of bacteria after 10 hours is given by n(10)=1000·2^{10/4} \approx 5657 bacteria. The number of bacteria after 24 hours is given by n(24)=1000·2^6=64,000 bacteria.

Given the exponential function f(x)=100·3^{x/2}, evaluate f(4) and f(10).

Solution

f(4)=900; \, f(10)=24,300.

Go to World Population Balance for another example of exponential population growth.

Graphing Exponential Functions

For any base b>0, \, b\ne 1″ height=”17″ width=”90″>, the exponential function <img loading= is defined for all real numbers x and b^x>0″ height=”13″ width=”49″>. Therefore, the domain of <img loading= is (−\infty ,\infty) and the range is (0,\infty). To graph b^x, we note that for b>1, \, b^x” height=”17″ width=”67″> is increasing on <img loading= and b^x \to \infty  as x \to \infty, whereas b^x \to 0 as x \to −\infty. On the other hand, if 0<b<1, \, f(x)=b^x is decreasing on (−\infty ,\infty) and b^x \to 0 as x \to \infty  whereas b^x \to \infty  as x \to −\infty  ((Figure)).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of four functions. The first function is “f(x) = 2 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing. The second function is “f(x) = 4 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing rapidly, more rapidly than the first function. The third function is “f(x) = (1/2) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. The third function is “f(x) = (1/4) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. It decrases at a faster rate than the third function.
Figure 2. If b>1″ height=”14″ width=”39″>, then <img loading= is increasing on (−\infty ,\infty). If 0<b<1, then b^x is decreasing on (−\infty ,\infty).

Visit this site for more exploration of the graphs of exponential functions.

Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.

Rule: Laws of Exponents

For any constants a>0, \, b>0″ height=”17″ width=”93″>, and for all <img loading= and y,

  1. b^x·b^y=b^{x+y}
  2. \large\frac{b^x}{b^y} \normalsize = b^{x-y}
  3. (b^x)^y=b^{xy}
  4. (ab)^x=a^x b^x
  5. \large \frac{a^x}{b^x} \normalsize =(\frac{a}{b})^x

Using the Laws of Exponents

Use the laws of exponents to simplify each of the following expressions.

  1. \large \frac{(2x^{2/3})^3}{(4x^{-1/3})^2}
  2. \large \frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}

Solution

  1. We can simplify as follows:\large \frac{(2x^{2/3})^3}{(4x^{-1/3})^2} \normalsize = \large \frac{2^3(x^{2/3})^3}{4^2(x^{-1/3})^2} \normalsize = \large \frac{8x^2}{16x^{-2/3}} \normalsize = \large \frac{x^2x^{2/3}}{2} \normalsize = \large \frac{x^{8/3}}{2}.
  2. We can simplify as follows:\large \frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \normalsize = \large \frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \normalsize = \large \frac{x^6y^{-2}}{x^{-2}y^{-4}} \normalsize = x^6x^2y^{-2}y^4 = x^8y^2.

Use the laws of exponents to simplify (6x^{-3}y^2)/(12x^{-4}y^5).

Solution

x/(2y^3)

Hint

x^a/x^b=x^{a-b}

The Number 

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests P dollars in a savings account with an annual interest rate r, compounded annually. The amount of money after 1 year isA(1)=P+rP=P(1+r).

The amount of money after 2 years isA(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)^2.

More generally, the amount after t years isA(t)=P(1+r)^t.

If the money is compounded 2 times per year, the amount of money after half a year isA(\frac{1}{2})=P+(\frac{r}{2})P=P(1+(\frac{r}{2})).

The amount of money after 1 year isA(1)=A(\frac{1}{2})+(\frac{r}{2})A(\frac{1}{2})=P(1+\frac{r}{2})+\frac{r}{2}(P(1+\frac{r}{2}))=P(1+\frac{r}{2})^2.

After t years, the amount of money in the account isA(t)=P(1+\frac{r}{2})^{2t}.

More generally, if the money is compounded n times per year, the amount of money in the account after t years is given by the functionA(t)=P(1+\frac{r}{n})^{nt}.

What happens as n\to \infty? To answer this question, we let m=n/r and write(1+\frac{r}{n})^{nt}=(1+\frac{1}{m})^{mrt},

and examine the behavior of (1+1/m)^m as m\to \infty, using a table of values ((Figure)).

\mathbf{m}10100100010,000100,0001,000,000
\mathbf{(1+\frac{1}{m})^m}2.59372.70482.716922.718152.7182682.718280

Looking at this table, it appears that (1+1/m)^m is approaching a number between 2.7 and 2.8 as m\to \infty . In fact, (1+1/m)^m does approach some number as m\to \infty . We call this number e. To six decimal places of accuracy,e \approx 2.718282.

The letter e was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between e and logarithmic functions. We still use the notation e today to honor Euler’s work because it appears in many areas of mathematics and because we can use it in many practical applications.

Returning to our savings account example, we can conclude that if a person puts P dollars in an account at an annual interest rate r, compounded continuously, then A(t)=Pe^{rt}. This function may be familiar. Since functions involving base e arise often in applications, we call the function f(x)=e^x the natural exponential function. Not only is this function interesting because of the definition of the number e, but also, as discussed next, its graph has an important property.

Since e>1″ height=”13″ width=”40″>, we know <img loading= is increasing on (−\infty ,\infty). In (Figure), we show a graph of f(x)=e^x along with a tangent line to the graph of at x=0. We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of f at x=a is a line that passes through the point (a,f(a)) and has the same “slope” as f at that point. The function f(x)=e^x is the only exponential function b^x with tangent line at x=0 that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the simplest exponential function to use in many instances.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of the function “f(x) = e to power of x”, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1). At this point, a line is drawn tangent to the function. This line has the label “slope = 1”.
Figure 3. The graph of f(x)=e^x has a tangent line with slope 1 at x=0.

Compounding Interest

Suppose \$500 is invested in an account at an annual interest rate of r=5.5\%, compounded continuously.

  1. Let t denote the number of years after the initial investment and A(t) denote the amount of money in the account at time t. Find a formula for A(t).
  2. Find the amount of money in the account after 10 years and after 20 years.

Solution

  1. If P dollars are invested in an account at an annual interest rate r, compounded continuously, then A(t)=Pe^{rt}. Here P=\$500 and r=0.055. Therefore, A(t)=500e^{0.055t}.
  2. After 10 years, the amount of money in the account isA(10)=500e^{0.055·10}=500e^{0.55}\approx \$866.63.After 20 years, the amount of money in the account isA(20)=500e^{0.055·20}=500e^{1.1}\approx \$1,502.08.

If \$750 is invested in an account at an annual interest rate of 4\%, compounded continuously, find a formula for the amount of money in the account after t years. Find the amount of money after 30 years.

[reveal-answer q=”505690″]Show Answer[/reveal-answer]
[hidden-answer a=”505690″]A(t)=750e^{0.04t}. After 30 years, there will be approximately \$2,490.09.

Hint

A(t)=Pe^{rt}.

Logarithmic Functions

Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.

The exponential function f(x)=b^x is one-to-one, with domain (−\infty ,\infty) and range (0,\infty ). Therefore, it has an inverse function, called the logarithmic function with base b. For any b>0, \, b \ne 1″ height=”17″ width=”90″>, the logarithmic function with base <img loading=, denoted \log_b, has domain (0,\infty ) and range (−\infty ,\infty ), and satisfies\log_b(x)=y if and only if b^y=x.

For example,

\begin{array}{cccc} \log_2 (8)=3\hfill & & & \text{since}\phantom{\rule{3em}{0ex}}2^3=8,\hfill \\ \log_{10} (\frac{1}{100})=-2\hfill & & & \text{since}\phantom{\rule{3em}{0ex}}10^{-2}=\frac{1}{10^2}=\frac{1}{100},\hfill \\ \log_b (1)=0\hfill & & & \text{since}\phantom{\rule{3em}{0ex}}b^0=1 \, \text{for any base} \, b>0.\hfill \end{array}

Furthermore, since y=\log_b (x) and y=b^x are inverse functions,\log_b (b^x)=x \, \text{and} \, b^{\log_b (x)}=x.

The most commonly used logarithmic function is the function \log_e (x). Since this function uses natural e as its base, it is called the natural logarithm. Here we use the notation \ln(x) or \ln x to mean \log_e (x). For example,\ln (e)=\log_e (e)=1, \, \ln(e^3)=\log_e (e^3)=3, \, \ln(1)=\log_e (1)=0.

Since the functions f(x)=e^x and g(x)=\ln(x) are inverses of each other,\ln(e^x)=x \, \text{and} \, e^{\ln x}=x,

and their graphs are symmetric about the line y=x ((Figure)).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 4. The graph is of two functions. The first function is “f(x) = e to power of x”, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1) and there is no x intercept. The second function is “f(x) = ln(x)”, an increasing curved function. The x intercept is at the point (1, 0) and there is no y intercept. A dotted line with label “y = x” is also plotted on the graph, to show that the functions are mirror images over this line.
Figure 4: The functions y=e^x and y=\ln(x) are inverses of each other, so their graphs are symmetric about the line y=x.

At this site you can see an example of a base-10 logarithmic scale.

In general, for any base b>0, \, b\ne 1″ height=”17″ width=”90″>, the function <img loading= is symmetric about the line y=x with the function f(x)=b^x. Using this fact and the graphs of the exponential functions, we graph functions \log_b (x) for several values of b>1″ height=”14″ width=”39″> (<a href=(Figure)).

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of three functions. All three functions a log functions that are increasing curved functions that start slightly to the right of the y axis and have an x intercept at (1, 0). The first function is “y = log base 10 (x)”, the second function is “f(x) = ln(x)”, and the third function is “y = log base 2 (x)”. The third function increases the most rapidly, the second function increases next most rapidly, and the third function increases the slowest.
Figure 5: Graphs of y=\log_b (x) are depicted for b=2, \, e, \, 10.

Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.

Rule: Properties of Logarithms

If a,b,c>0, \, b\ne 1″ height=”17″ width=”123″>, and <img loading= is any real number, then

\begin{array}{cccc}1.\phantom{\rule{2em}{0ex}}\log_b (ac)=\log_b (a)+\log_b (c)\hfill & & & \text{(Product property)}\hfill \\ 2.\phantom{\rule{2em}{0ex}}\log_b(\frac{a}{c})=\log_b (a) -\log_b (c)\hfill & & & \text{(Quotient property)}\hfill \\ 3.\phantom{\rule{2em}{0ex}}\log_b (a^r)=r \log_b (a)\hfill & & & \text{(Power property)}\hfill \end{array}

Solving Equations Involving Exponential Functions

Solve each of the following equations for x.

  1. 5^x=2
  2. e^x+6e^{−x}=5

Solution

  1. Applying the natural logarithm function to both sides of the equation, we have\ln 5^x=\ln 2.Using the power property of logarithms,x \ln 5=\ln 2.Therefore, x=\ln 2 / \ln 5.
  2. Multiplying both sides of the equation by e^x, we arrive at the equatione^{2x}+6=5e^x.Rewriting this equation ase^{2x}-5e^x+6=0,we can then rewrite it as a quadratic equation in e^x:(e^x)^2-5(e^x)+6=0.Now we can solve the quadratic equation. Factoring this equation, we obtain(e^x-3)(e^x-2)=0.Therefore, the solutions satisfy e^x=3 and e^x=2. Taking the natural logarithm of both sides gives us the solutions x=\ln 3, \, \ln 2.

Solve e^{2x} / (3+e^{2x})=1/2.

Solution

x=\frac{\ln 3}{2}

Hint

First solve the equation for e^{2x}.

Solving Equations Involving Logarithmic Functions

Solve each of the following equations for x.

  1. \ln (\frac{1}{x})=4
  2. \log_{10} \sqrt{x}+ \log_{10} x=2
  3. \ln(2x)-3 \ln(x^2)=0

Solution

  1. By the definition of the natural logarithm function,\ln\big(\frac{1}{x}\big)=4 \, \text{if and only if} \, e^4=\frac{1}{x}.Therefore, the solution is x=1/e^4.
  2. Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as\log_{10} \sqrt{x}+ \log_{10} x = \log_{10} x \sqrt{x} = \log_{10}x^{3/2} = \frac{3}{2} \log_{10} x.Therefore, the equation can be rewritten as\frac{3}{2} \log_{10} x = 2 \, \text{or} \, \log_{10} x = \frac{4}{3}.The solution is x=10^{4/3}=10\sqrt[3]{10}.
  3. Using the power property of logarithmic functions, we can rewrite the equation as \ln(2x) - \ln(x^6) = 0.
    Using the quotient property, this becomes\ln\big(\frac{2}{x^5}\big)=0.Therefore, 2/x^5=1, which implies x=\sqrt[5]{2}. We should then check for any extraneous solutions.

Solve \ln(x^3)-4 \ln (x)=1.

Solution

x=\frac{1}{e}

Hint

First use the power property, then use the product property of logarithms.

When evaluating a logarithmic function with a calculator, you may have noticed that the only options are \log_{10} or log, called the common logarithm, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base b. If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.

Rule: Change-of-Base Formulas

Let a>0, \, b>0″ height=”17″ width=”93″>, and <img loading=.

  1. a^x=b^{x \log_b a} for any real number x.
    If b=e, this equation reduces to a^x=e^{x \log_e a}=e^{x \ln a}.
  2. \log_a x=\frac{\log_b x}{\log_b a} for any real number x>0″ height=”12″ width=”43″>.<br>If <img loading=, this equation reduces to \log_a x=\frac{\ln x}{\ln a}.

Proof

For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base b>0, \, b\ne 1, \, \log_b (a^x)=x \log_b a” height=”18″ width=”241″>. Therefore,<img loading=.

In addition, we know that b^x and \log_b (x) are inverse functions. Therefore,b^{\log_b (a^x)}=a^x.

Combining these last two equalities, we conclude that a^x=b^{x \log_b a}.

To prove the second property, we show that(\log_b a)·(\log_a x)=\log_b x.

Let u=\log_b a, \, v=\log_a x, and w=\log_b x. We will show that u·v=w. By the definition of logarithmic functions, we know that b^u=a, \, a^v=x, and b^w=x. From the previous equations, we see thatb^{uv}=(b^u)^v=a^v=x=b^w.

Therefore, b^{uv}=b^w. Since exponential functions are one-to-one, we can conclude that u·v=w.

Changing Bases

Use a calculating utility to evaluate \log_3 7 with the change-of-base formula presented earlier.

Solution

Use the second equation with a=3 and e=3:

\log_3 7=\frac{\ln 7}{\ln 3} \approx 1.77124.

Use the change-of-base formula and a calculating utility to evaluate \log_4 6.

Solution

1.29248

Hint

Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.

Chapter Opener: The Richter Scale for Earthquakes

A photograph of an earthquake fault.
Figure 6: (credit: modification of work by Robb Hannawacker, NPS)

In 1935, Charles Richter developed a scale (now known as the Richter scale) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude R_1 on the Richter scale and a second earthquake with magnitude R_2 on the Richter scale. Suppose R_1 > R_2″ height=”16″ width=”65″>, which means the earthquake of magnitude <img loading= is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If A_1 is the amplitude measured for the first earthquake and A_2 is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:R_1 - R_2 = \log_{10}(\frac{A_1}{A_2}).

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,8-7=\log_{10}(\frac{A_1}{A_2}).

Therefore,\log_{10}(\frac{A_1}{A_2})=1,

which implies A_1 / A_2 = 10 or A_1 = 10A_2. Since A_1 is 10 times the size of A_2, we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation\log_{10}(\frac{A_1}{A_2})=8-6=2.

Therefore, A_1=100A_2. That is, the first earthquake is 100 times more intense than the second earthquake.

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?

Solution

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:

9-7.3=\log_{10}(\frac{A_1}{A_2}).

Therefore, A_1 / A_2=10^{1.7}, and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.

Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.

The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.

Hint

R_1-R_2=\log_{10}(A_1 / A_2).

Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of e^x and e^{−x}. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ((Figure)). If we introduce a coordinate system so that the low point of the chain lies along the y-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.

A photograph of a spider web collecting dew drops.
Figure 7. The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: “Mtpaley”, Wikimedia Commons)

Definition

Hyperbolic cosine

\cosh x=\large \frac{e^x+e^{−x}}{2}

Hyperbolic sine

\sinh x=\large \frac{e^x-e^{−x}}{2}

Hyperbolic tangent

\tanh x=\large \frac{\sinh x}{\cosh x} \normalsize = \large \frac{e^x-e^{−x}}{e^x+e^{−x}}

Hyperbolic cosecant

\text{csch} \, x=\large \frac{1}{\sinh x} \normalsize = \large \frac{2}{e^x-e^{−x}}

Hyperbolic secant

\text{sech} \, x=\large \frac{1}{\cosh x} \normalsize = \large \frac{2}{e^x+e^{−x}}

Hyperbolic cotangent

\coth x=\large \frac{\cosh x}{\sinh x} \normalsize = \large \frac{e^x+e^{−x}}{e^x-e^{−x}}

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanhsechcsch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

Using the definition of \cosh(x) and principles of physics, it can be shown that the height of a hanging chain, such as the one in (Figure), can be described by the function h(x)=a \cosh(x/a)+c for certain constants a and c.

But why are these functions called hyperbolic functions? To answer this question, consider the quantity \cosh^2 t-\sinh^2 t. Using the definition of \cosh and \sinh, we see that\cosh^2 t-\sinh^2 t=\large \frac{e^{2t}+2+e^{-2t}}{4}-\frac{e^{2t}-2+e^{-2t}}{4} \normalsize =1.

This identity is the analog of the trigonometric identity \cos^2 t+\sin^2 t=1. Here, given a value t, the point (x,y)=(\cosh t,\sinh t) lies on the unit hyperbola x^2-y^2=1 ((Figure)).

An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation “(x squared) - (y squared) -1”. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways “U” shape. There is a point plotted on the graph of the relation labeled “(cosh(1), sinh(1))”, which is at the approximate point (1.5, 1.2).
Figure 8.The unit hyperbola \cosh^2 t-\sinh^2 t=1.

Graphs of Hyperbolic Functions

To graph \cosh x and \sinh x, we make use of the fact that both functions approach (1/2)e^x as x \to \infty, since e^{−x} \to 0 as x \to \infty. As x \to −\infty, \, \cosh x approaches 1/2e^{−x}, whereas \sinh x approaches -1/2e^{−x}. Therefore, using the graphs of 1/2e^x, \, 1/2e^{−x}, and −1/2e^{−x} as guides, we graph \cosh x and \sinh x. To graph \tanh x, we use the fact that \tanh(0)=1, \, -1<\tanh(x)<1 for all x, \, \tanh x \to 1 as x \to \infty, and \tanh x \to −1 as x \to −\infty. The graphs of the other three hyperbolic functions can be sketched using the graphs of \cosh x, \, \sinh x, and \tanh x ((Figure)).

An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function “y = cosh(x)”, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of -x)”, a decreasing curved function and the second of these functions is “y = (1/2)(e to power of x)”, an increasing curved function. The function “y = cosh(x)” is always above these two functions without ever touching them. The second graph is of the function “y = sinh(x)”, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of x)”, an increasing curved function and the second of these functions is “y = -(1/2)(e to power of -x)”, an increasing curved function that approaches the x axis without touching it. The function “y = sinh(x)” is always between these two functions without ever touching them. The third graph is of the function “y = sech(x)”, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function “y = csch(x)”. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function “y = tanh(x)”, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = 1”, a horizontal line function and the second of these functions is “y = -1”, another horizontal line function. The function “y = tanh(x)” is always between these two functions without ever touching them. The sixth graph is of the function “y = coth(x)”. On the left side of the y axis, the function starts slightly below the boundary line “y = 1” and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line “y = -1”, which it never touches.
Figure 9. The hyperbolic functions involve combinations of e^x and e^{−x}.

Identities Involving Hyperbolic Functions

The identity \cosh^2 t-\sinh^2 t, shown in (Figure), is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.

Rule: Identities Involving Hyperbolic Functions

  1. \cosh(−x)=\cosh x
  2. \sinh(−x)=−\sinh x
  3. \cosh x+\sinh x=e^x
  4. \cosh x-\sinh x=e^{−x}
  5. \cosh^2 x-\sinh^2 x=1
  6. 1-\tanh^2 x=\text{sech}^2 x
  7. \coth^2 x-1=\text{csch}^2 x
  8. \sinh(x \pm y)=\sinh x \cosh y \pm \cosh x \sinh y
  9. \cosh (x \pm y)=\cosh x \cosh y \pm \sinh x \sinh y

Evaluating Hyperbolic Functions

  1. Simplify \sinh(5 \ln x).
  2. If \sinh x=3/4, find the values of the remaining five hyperbolic functions.

Solution

  1. Using the definition of the \sinh function, we write\sinh(5 \ln x)=\large \frac{e^{5 \ln x}-e^{-5 \ln x}}{2} \normalsize = \large \frac{e^{\ln(x^5)}-e^{\ln(x^{-5})}}{2} \normalsize =\large \frac{x^5-x^{-5}}{2}.
  2. Using the identity \cosh^2 x-\sinh^2 x=1, we see that\cosh^2 x=1+\big(\frac{3}{4}\big)^2=\frac{25}{16}.Since \cosh x \ge 1 for all x, we must have \cosh x=5/4. Then, using the definitions for the other hyperbolic functions, we conclude that \tanh x=3/5, \, \text{csch} \, x=4/3, \, \text{sech} \, x=4/5, and \coth x=5/3.

Simplify \cosh(2 \ln x).

Solution

(x^2+x^{-2})/2

Hint

Use the definition of the cosh function and the power property of logarithm functions.

Inverse Hyperbolic Functions

From the graphs of the hyperbolic functions, we see that all of them are one-to-one except \cosh x and \text{sech} \, x. If we restrict the domains of these two functions to the interval [0,\infty), then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.

Definition

Inverse Hyperbolic Functions

\begin{array}{cccc}\sinh^{-1} x=\text{arcsinh } x=\ln(x+\sqrt{x^2+1})\hfill & & & \cosh^{-1} x=\text{arccosh } x=\ln(x+\sqrt{x^2-1})\hfill \\ \tanh^{-1} x=\text{arctanh } x=\frac{1}{2}\ln\big(\frac{1+x}{1-x}\big)\hfill & & & \coth^{-1} x=\text{arccot } x=\frac{1}{2}\ln\big(\frac{x+1}{x-1}\big)\hfill \\ \text{sech}^{-1} x=\text{arcsech } x=\ln\big(\frac{1+\sqrt{1-x^2}}{x}\big)\hfill & & & \text{csch}^{-1} x=\text{arccsch } x=\ln\big(\frac{1}{x}+\frac{\sqrt{1+x^2}}{|x|}\big)\hfill \end{array}

Let’s look at how to derive the first equation. The others follow similarly. Suppose y=\sinh^{-1} x. Then, x=\sinh y and, by the definition of the hyperbolic sine function, x=\frac{e^y-e^{−y}}{2}. Therefore,e^y-2x-e^{−y}=0.

Multiplying this equation by e^y, we obtaine^{2y}-2xe^y-1=0.

This can be solved like a quadratic equation, with the solutione^y=\large \frac{2x \pm \sqrt{4x^2+4}}{2} \normalsize =x \pm \sqrt{x^2+1}.

Since e^y>0″ height=”12″ width=”49″>, the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that<img loading=.

Evaluating Inverse Hyperbolic Functions

Evaluate each of the following expressions.

\sinh^{-1}(2)
\tanh^{-1}(1/4)

Solution

\sinh^{-1}(2)=\ln(2+\sqrt{2^2+1})=\ln(2+\sqrt{5}) \approx 1.4436
\tanh^{-1}(1/4)=\frac{1}{2}\ln(\frac{1+1/4}{1-1/4})=\frac{1}{2}\ln(\frac{5/4}{3/4})=\frac{1}{2}\ln(\frac{5}{3}) \approx 0.2554

Evaluate \tanh^{-1}(1/2).

Solution

\frac{1}{2}\ln(3) \approx 0.5493.

Hint

Use the definition of \tanh^{-1} x and simplify.

Key Concepts

  • The exponential function y=b^x is increasing if b>1″ height=”14″ width=”39″> and decreasing if <img loading=. Its domain is (−\infty ,\infty) and its range is (0,\infty).
  • The logarithmic function y=\log_b(x) is the inverse of y=b^x. Its domain is (0,\infty) and its range is (−\infty,\infty).
  • The natural exponential function is y=e^x and the natural logarithmic function is y=\ln x=\log_e x.
  • Given an exponential function or logarithmic function in base a, we can make a change of base to convert this function to any base b>0, \, b \ne 1″ height=”17″ width=”90″>. We typically convert to base <img loading=.
  • The hyperbolic functions involve combinations of the exponential functions e^x and e^{−x}. As a result, the inverse hyperbolic functions involve the natural logarithm.