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

Its population after 2 years is

In general, its population after $t$ years is

which is an exponential function. More generally, any function of the form $f\left(x\right)=b^x,$ where $b>0,b\ne 1,$ 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 $f\left(x\right)=x^b$ 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 [link] , 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 $x\to \text{−}\infty ,x^2\to \infty ,$ whereas $2^x\to 0.$ The line $y=0$ is a horizontal asymptote for $y=2^x.$

In [link] , we graph both $y=x^2$ and $y=2^x$ to show how the graphs differ.

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 $x=\text{−}y$ for some positive integer $y,$ and we define $b^x=b^{\text{−}y}=1\text{/}{b}^y.$ Also, $b^0$ is defined to be $1.$ If $x$ is a rational number, then $x=p\text{/}q,$ where $p$ and $q$ are integers and $b^x=b^{p\text{/}q}=\sqrt[q]{b^p}.$ For example, $9^{3\text{/}2}=\sqrt{9^3}=27.$ 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 [link] , 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 $2^{\sqrt{2}}.$