1.5 Exponential and logarithmic functions  (Page 4/17)

The exponential function $f\left(x\right)=b^x$ is one-to-one, with domain $(\text{−}\infty ,\infty )$ and range $\left(0,\infty \right).$ Therefore, it has an inverse function, called the logarithmic function with base $b.$ For any $b>0,b\ne 1,$ the logarithmic function with base b , denoted ${\text{log}}_b,$ has domain $(0,\infty )$ and range $\left(\text{−}\infty ,\infty \right),$ and satisfies

For example,

Furthermore, since $y={\text{log}}_b(x)$ and $y=b^x$ are inverse functions,

The most commonly used logarithmic function is the function ${\text{log}}_e.$ Since this function uses natural $e$ as its base, it is called the natural logarithm    . Here we use the notation $\text{ln}(x)$ or $\text{ln}x$ to mean ${\text{log}}_e\left(x\right).$ For example,

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

and their graphs are symmetric about the line $y=x$ ( [link] ).

In general, for any base $b>0,b\ne 1,$ the function $g\left(x\right)={\text{log}}_b(x)$ is symmetric about the line $y=x$ with the function $f\left(x\right)=b^x.$ Using this fact and the graphs of the exponential functions, we graph functions ${\text{log}}_b$ for several values of $b>1$ ( [link] ).

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