1.5 Exponential and Logarithmic Functions

In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number $e .$ We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations, and prove that the functions have the same properties with either definition.)

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 is

P (1) = P_{0} + 0.02 P_{0} = P_{0} (1 + 0.02) = P_{0} (1.02) .

Its population after 2 years is

P (2) = P (1) + 0.02 P (1) = P (1) (1.02) = P_{0} (1.02)^{2} .

In general, its population after $t$ years is

P (t) = P_{0} (1.02)^{t},

which is an exponential function. More generally, any function of the form $f (x) = b^{x},$ where $b > 0, b \neq = 1,$ is an exponential function with base $b$ 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 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 Table 1.10, we see that both $2^{x}$ and $x^{2}$ approach infinity as Eventually, however, becomes larger than and grows more rapidly as In the opposite direction, as whereas The line is a horizontal asymptote for

Table 1.10Values of

x^{2}

and

2^{x}

x

- 3

- 2

- 1

0

In Figure 1.43, 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 1.43Both $2^{x}$ and $x^{2}$ approach infinity as $x \to \infty,$ but $2^{x}$ grows more rapidly than As whereas

Evaluating Exponential Functions

Recall the properties of exponents: If $x$ is a positive integer, then we define $b^{x} = b \cdot b \dots b$ (with $x$ factors of $b) .$ If $x$ is a negative integer, then $x = - y$ for some positive integer and we define Also, is defined to be If is a rational number, then where and are integers and For example, However, how is defined if is an irrational number? For example, what do we mean by This is too complex a question for us to answer fully right now; however, we can make an approximation. In Table 1.11, we list some rational numbers approaching and the values of for each rational number are presented as well. We claim that if we choose rational numbers getting closer and closer to the values of get closer and closer to some number We define that number to be

Table 1.11Values of

2^{x}

for a List of Rational Numbers Approximating

2

x

1.4

Graphing Exponential Functions

For any base $b > 0, b \neq = 1,$ the exponential function $f (x) = b^{x}$ is defined for all real numbers $x$ and $b^{x} > 0.$ Therefore, the domain of is and the range is To graph we note that for is increasing on and as whereas as On the other hand, if is decreasing on and as whereas as (Figure 1.44).

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 decreases at a faster rate than the third function. — Figure 1.44If $b > 1,$ then $b^{x}$ is increasing on $(- \infty, \infty) .$ If $0 < b < 1,$ then is decreasing on

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

The Number e

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 is

A (1) = P + r P = P (1 + r) .

The amount of money after $2$ years is

A (2) = A (1) + r A (1) = P (1 + r) + r P (1 + r) = P (1 + r)^{2} .

More generally, the amount after $t$ years is

A (t) = P (1 + r)^{t} .

If the money is compounded 2 times per year, the amount of money after half a year is

A (\frac{1}{2}) = P + (\frac{r}{2}) P = P (1 + (\frac{r}{2})) .

The amount of money after $1$ year is

A (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 is

A (t) = P (1 + \frac{r}{2})^{2 t} .

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 function

A (t) = P (1 + \frac{r}{n})^{n t} .

What happens as $n \to \infty ?$ To answer this question, we let $m = n / r$ and write

(1 + \frac{r}{n})^{n t} = (1 + \frac{1}{m})^{m r t},

and examine the behavior of $(1 + 1 / m)^{m}$ as $m \to \infty,$ using a table of values (Table 1.12).

Table 1.12Values of

(1 + \frac{1}{m})^{m}

m \to \infty

m

10

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 . 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 $A (t) = P e^{r t} .$ This function may be familiar. Since functions involving base $e$ arise often in applications, we call the function $f (x) = e^{x}$ the . Not only is this function interesting because of the definition of the number but also, as discussed next, its graph has an important property.

Since $e > 1,$ we know $e^{x}$ is increasing on $(- \infty, \infty) .$ In Figure 1.45, 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 at is a line that passes through the point and has the same “slope” as at that point The function is the only exponential function with tangent line at that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the most simple 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 1.45The graph of $f (x) = e^{x}$ has a tangent line with slope $1$ at $x = 0.$

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 $b > 0, b \neq = 1,$ the logarithmic function with base , denoted has domain and range and satisfies

lo g_{b} (x) = y if and only if b^{y} = x .

For example,

lo g_{2} (8) = 3 lo g_{10} (\frac{1}{100}) = - 2 lo g_{b} (1) = 0 since 2^{3} = 8, since 10^{- 2} = \frac{1}{10 ^{2}} = \frac{1}{100}, since b^{0} = 1 for any base b > 0.

Furthermore, since $y = lo g_{b} (x)$ and $y = b^{x}$ are inverse functions,

lo g_{b} (b^{x}) = x and b^{l o g_{b} (x)} = x .

The most commonly used logarithmic function is the function $lo g_{e} .$ 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 $lo g_{e} (x) .$ For example,

ln (e) = lo g_{e} (e) = 1, ln (e^{3}) = lo g_{e} (e^{3}) = 3, ln (1) = lo g_{e} (1) = 0.

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

ln (e^{x}) = x and e^{l n x} = x,

and their graphs are symmetric about the line $y = x$ (Figure 1.46).

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 1.46The functions $y = e^{x}$ and $y = ln (x)$ are inverses of each other, so their graphs are symmetric about the line $y = x .$

In general, for any base $b > 0, b \neq = 1,$ the function $g (x) = lo g_{b} (x)$ is symmetric about the line $y = x$ with the function Using this fact and the graphs of the exponential functions, we graph functions for several values of (Figure 1.47).

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 1.47Graphs of $y = lo g_{b} (x)$ are depicted for $b = 2, e, 10.$

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

When evaluating a logarithmic function with a calculator, you may have noticed that the only options are $lo g_{10}$ 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.

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 $b > 0, b \neq = 1, lo g_{b} (a^{x}) = x lo g_{b} a .$ Therefore,

b^{l o g_{b} (a^{x})} = b^{x l o g_{b} a} .

In addition, we know that $b^{x}$ and $lo g_{b} (x)$ are inverse functions. Therefore,

b^{l o g_{b} (a^{x})} = a^{x} .

Combining these last two equalities, we conclude that $a^{x} = b^{x l o g_{b} a} .$

To prove the second property, we show that

(lo g_{b} a) \cdot (lo g_{a} x) = lo g_{b} x .

Let $u = lo g_{b} a, v = lo g_{a} x,$ and $w = lo g_{b} x .$ We will show that By the definition of logarithmic functions, we know that and From the previous equations, we see that

b^{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 $u \cdot v = w .$

□

Example 1.39Chapter Opener: The Richter Scale for Earthquakes

Figure 1.48(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 $R_{1} > R_{2},$ which means the earthquake of magnitude 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 is the amplitude measured for the first earthquake and is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:

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 1.49). 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 1.49The 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)

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh, sech, csch, 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 1.49, 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 = \frac{e ^{2 t} + 2 + e ^{- 2 t}}{4} - \frac{e ^{2 t} - 2 + e ^{- 2 t}}{4} = 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 (Figure 1.50).

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 1.50The 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 As approaches whereas approaches Therefore, using the graphs of and as guides, we graph and To graph we use the fact that for all as and as The graphs of the other three hyperbolic functions can be sketched using the graphs of and (Figure 1.51).

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 1.51The hyperbolic functions involve combinations of $e^{x}$ and

Identities Involving Hyperbolic Functions

The identity $cosh^{2} t - sinh^{2} t,$ shown in Figure 1.50, 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.

Inverse Hyperbolic Functions

From the graphs of the hyperbolic functions, we see that all of them are one-to-one except $cosh x$ and $sech x .$ 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.

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} - 2 x - e^{- y} = 0.

Multiplying this equation by $e^{y},$ we obtain

e^{2 y} - 2 x e^{y} - 1 = 0.

This can be solved like a quadratic equation, with the solution

e^{y} = \frac{2 x \pm 4 x ^{2} + 4}{2} = x \pm x^{2} + 1 .

Since $e^{y} > 0,$ the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that

y = ln (x + x^{2} + 1) .

Key Concepts

The exponential function $y = b^{x}$ is increasing if $b > 1$ and decreasing if $0 < b < 1.$ Its domain is $(- \infty, \infty)$ and its range is $(0, \infty) .$