1.5 Exponential and logarithmic functions  (Page 2/17)

Graphing exponential functions

For any base $b>0,b\ne 1,$ the exponential function $f(x)=b^x$ is defined for all real numbers $x$ and $b^x>0.$ Therefore, the domain of $f\left(x\right)=b^x$ is $\left(\text{−}\infty ,\infty \right)$ and the range is $\left(0,\infty \right).$ To graph $b^x,$ we note that for $b>1,b^x$ is increasing on $(\text{−}\infty ,\infty )$ and $b^x\to \infty$ as $x\to \infty ,$ whereas $b^x\to 0$ as $x\to \text{−}\infty .$ On the other hand, if $0<b<1,f\left(x\right)=b^x$ is decreasing on $(\text{−}\infty ,\infty )$ and $b^x\to 0$ as $x\to \infty$ whereas $b^x\to \infty$ as $x\to \text{−}\infty$ ( [link] ).

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

The amount of money after $2$ years is

More generally, the amount after $t$ years is

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

The amount of money after $1$ year is

After $t$ years, the amount of money in the account is

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