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 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
which is an exponential function. More generally, any function of the form \(𝑓(𝑥)=𝑏^𝑥\text{,}\) where 𝑏>0,𝑏≠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 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 \(𝑦=2^𝑥\text{.}\) In Table 1.5.1, we see that both \(2^𝑥\) and \(𝑥^2\) approach infinity as \(𝑥\to\infty\text{.}\) Eventually, however, \(2^𝑥\) becomes larger than \(𝑥^2\) and grows more rapidly as \(𝑥\to\infty\text{.}\) In the opposite direction, as \(𝑥\to-\infty\text{,}\)\(𝑥^2\to\infty\text{,}\) whereas \(2^𝑥\to0\text{.}\) The line \(𝑦=0\) is a horizontal asymptote for \(𝑦=2^𝑥\text{.}\)
Figure1.5.2.Both \(2^𝑥\) and \(𝑥^2\) approach infinity as \(𝑥\to\infty\text{,}\) but \(2^𝑥\) grows more rapidly than \(𝑥^2\text{.}\) As \(𝑥\to-\infty\text{,}\)\(𝑥^2\to\infty\text{,}\) whereas \(2^𝑥\to0\text{.}\)
Recall the properties of exponents: If 𝑥 is a positive integer, then we define \(𝑏^𝑥=𝑏·𝑏⋯𝑏\) (with \(𝑥\) factors of \(𝑏\)). If 𝑥 is a negative integer, then 𝑥=−𝑦 for some positive integer 𝑦, and we define \(𝑏^𝑥=𝑏^{−𝑦}=1/𝑏^𝑦\text{.}\) Also, \(𝑏^0\) is defined to be 1. If 𝑥 is a rational number, then \(𝑥=𝑝/𝑞\text{,}\) where 𝑝 and 𝑞 are integers and \(𝑏^𝑥=𝑏^(𝑝/𝑞)=𝑞√𝑏^𝑝\text{.}\) For example, \(9^{3/2}=√9^3=27\text{.}\) However, how is \(𝑏^𝑥\) defined if 𝑥 is an irrational number? For example, what do we mean by 2^√2? This is too complex a question for us to answer fully right now; however, we can make an approximation. In Table 1.5.3, we list some rational numbers approaching √2, and the values of \(2𝑥\) for each rational number 𝑥 are presented as well. We claim that if we choose rational numbers 𝑥 getting closer and closer to √2, the values of \(2𝑥\) get closer and closer to some number \(𝐿\text{.}\) We define that number \(𝐿\) to be 2√2.
Table1.5.3.Values of \(2^𝑥\) for a List of Rational Numbers Approximating √2.
For any base 𝑏>0,𝑏≠1, the exponential function \(𝑓(𝑥)=𝑏^𝑥\) is defined for all real numbers 𝑥 and \(𝑏^𝑥>0\text{.}\) Therefore, the domain of 𝑓\((𝑥)=𝑏^𝑥\) is \((-\infty,\infty)\) and the range is \((0,\infty)\text{.}\) To graph \(𝑏^𝑥\text{,}\) we note that for \(𝑏>1,𝑏^𝑥\) is increasing on \((-\infty,\infty)\) and \(𝑏^𝑥\to\infty\) as \(𝑥\to\infty\text{,}\) whereas \(𝑏^𝑥\to0\) as \(𝑥\to−\infty\text{.}\) On the other hand, if 0<𝑏<1,\(𝑓(𝑥)=𝑏^𝑥\) is decreasing on \((-\infty,\infty)\) and \(𝑏^𝑥\to0\) as \(𝑥\to\infty\) whereas \(𝑏^𝑥\to\infty\) as \(𝑥\to−\infty\) (Figure 1.5.4).
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
Looking at this table, it appears that \((1+1/𝑚)^𝑚\) is approaching a number between 2.7 and 2.8 as \(𝑚\to\infty\text{.}\) In fact, \((1+1/𝑚)^𝑚\) does approach some number as \(𝑚\to\infty\text{.}\) We call this number 𝑒. To six decimal places of accuracy, 𝑒≈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 \(𝐴(𝑡)=𝑃𝑒^{𝑟𝑡}\text{.}\) This function may be familiar. Since functions involving base 𝑒 arise often in applications, we call the function \(𝑓(𝑥)=𝑒𝑥\) 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, we know \(𝑒^𝑥\) is increasing on \((-\infty,\infty)\text{.}\) In Figure 1.5.6, 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 most simple exponential function to use in many instances.
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 \((0,\infty)\text{.}\) Therefore, it has an inverse function, called the logarithmic function with base 𝑏. For any 𝑏>0,𝑏≠1, the logarithmic function with base b, denoted \(\log_𝑏\text{,}\) has domain \((0,\infty)\) and range \((-\infty,\infty)\text{,}\) and satisfies
\begin{equation*}
\log_𝑏(𝑥)=𝑦\text{ if and only if }𝑏^𝑦=𝑥.
\end{equation*}
For example, \(\log_2(8)=3\) since \(2^3=8\text{,}\)\(\log_{10}(1/100)=−2\) since \(10^{−2}=1/10^2=1/100\text{,}\)\(\log_𝑏(1)=0\) since \(𝑏^0=1\) for any base \(𝑏>0\text{.}\)
Furthermore, since \(𝑦=\log_𝑏(𝑥)\) and \(𝑦=𝑏^𝑥\) are inverse functions, \(\log_𝑏(𝑏^𝑥)=𝑥\) and \(𝑏^{\log_𝑏(𝑥)}=𝑥\text{.}\) The most commonly used logarithmic function is the function \(\log_𝑒\text{.}\) Since this function uses natural 𝑒 as its base, it is called the natural logarithm. Here we use the notation \(ln(𝑥)\) or \(ln𝑥\) to mean \(\log_𝑒(𝑥)\text{.}\) For example, \(ln(𝑒)=\log_𝑒(𝑒)=1,ln(𝑒^3)=\log_𝑒(𝑒^3)=3,ln(1)=\log_𝑒(1)=0\text{.}\) Since the functions \(𝑓(𝑥)=𝑒^𝑥\) and \(𝑔(𝑥)=ln(𝑥)\) are inverses of each other, \(ln(𝑒^𝑥)=𝑥\) and \(𝑒^{ln𝑥}=𝑥\text{,}\) and their graphs are symmetric about the line 𝑦=𝑥 (Figure 1.5.7).
In general, for any base \(𝑏>0,𝑏\neq1\text{,}\) the function \(𝑔(𝑥)=\log_𝑏(𝑥)\) is symmetric about the line 𝑦=𝑥 with the function \(𝑓(𝑥)=𝑏^𝑥\text{.}\) Using this fact and the graphs of the exponential functions, we graph functions \(\log_𝑏\) for several values of \(𝑏>1\) (Figure 1.5.8).
When evaluating a logarithmic function with a calculator, you may have noticed that the only options are \(\log_{10}\) or \(log\text{,}\) 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 \(𝑏\text{.}\) 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.
Let \(𝑎>0,𝑏>0\) with \(𝑎\neq1,𝑏\neq1\text{.}\) Then \(𝑎^𝑥=𝑏^{𝑥\log_𝑏𝑎}\) for any real number \(𝑥\text{.}\) If \(𝑏=𝑒\text{,}\) this equation reduces to \(𝑎^𝑥=𝑒^𝑥\log_𝑒𝑎=𝑒^𝑥ln𝑎\text{.}\)
Equivalently, \(\log_𝑎𝑥=\log_𝑏𝑥\cdot\log_𝑏𝑎\) for any real number \(𝑥>0\text{.}\) If \(𝑏=𝑒\text{,}\) this equation reduces to \(\log_𝑎 𝑥=\ln 𝑥 \ln 𝑎\) .
Figure1.5.11.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)
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 1.5.11, can be described by the function ℎ\((𝑥)=𝑎cosh(𝑥/𝑎)+𝑐\) for certain constants 𝑎 and 𝑐.
But why are these functions called hyperbolic functions? To answer this question, consider the quantity \(\cosh 2𝑡−\sinh 2𝑡\text{.}\) Using the definition of \(\cosh\) and \(\sinh\text{,}\) we see that
This identity is the analog of the trigonometric identity \(\cos^2𝑡+\sin^2𝑡=1\text{.}\) Here, given a value \(𝑡\text{,}\) the point \((𝑥,𝑦)=(\cosh 𝑡,\sinh 𝑡)\) lies on the unit hyperbola \(𝑥^2−𝑦^2=1\) (Figure 1.5.13).
To graph \(\cosh 𝑥\) and \(\sinh 𝑥\text{,}\) we make use of the fact that both functions approach \((1/2)𝑒^𝑥\) as \(𝑥\to\infty\text{,}\) since \(𝑒^{−𝑥}\to0\) as \(𝑥\to\infty\text{.}\) As \(x\to-\infty\text{,}\)\(\cosh 𝑥\) approaches \(1/2𝑒^{−𝑥}\text{,}\) whereas \(\sinh 𝑥\) approaches \(−1/2𝑒^{−𝑥}\text{.}\) Therefore, using the graphs of \(1/2𝑒^𝑥,1/2𝑒^{−𝑥}\text{,}\) and \(−1/2𝑒^{−𝑥}\) as guides, we graph \(\cosh 𝑥\) and \(\sinh 𝑥\text{.}\)
To graph \(tanh 𝑥\text{,}\) we use the fact that \(tanh(0)=0,−1<tanh(𝑥)<1\) for all \(𝑥\text{,}\)\(tanh 𝑥\to1\) as \(𝑥\to\infty\text{,}\) and \(tanh 𝑥\to−1\) as \(x\to-\infty\text{.}\) The graphs of the other three hyperbolic functions can be sketched using the graphs of \(\cosh 𝑥,\sinh 𝑥\text{,}\) and \(tanh 𝑥\) (Figure 1.5.14).
