Natural Logarithmic Function

If you invest $1000 at 5% interest compounded continuously, how long will it take for you to be a millionaire? Here we will look at:

• the definition of the natural logarithmic function and its relation to the natural exponential function,

• how to graph the natural logarithmic function, and

• how to convert a logarithmic function to a natural logarithmic function.

Natural Logarithmic Function Definition

Remember that e is the base used in the exponential growth and decay function $g\left(x\right)=e^x$. For more details see Exponential Growth and Decay. In addition, you know that exponential functions and logarithms are inverses of each other, so the inverse of the exponential growth function is $f\left(x\right)={\log }_e\left(x\right)$. However, this natural logarithm gets used so much that it has a shorthand:

The graph below shows the natural log is the reflection of the exponential growth function over the line$y=x$.

The natural log and the exponential growth function | StudySmarter Originals

In intuitive terms, the exponential function tells you how much something has grown given an amount of time, and the natural log gives you the amount of time it takes to reach a certain amount of growth. You can think of it as

$$\begin{gathered} e^{time}=howmuchgrowth \\ \ln (howmuchgrowth)=time \end{gathered}$$

The Domain of the Natural Logarithmic Function

Properties of the natural logarithmic function

Because the natural logarithmic function is just a logarithm base e, it has the same properties as the regular logarithmic function.

Properties of the natural logarithmic function:

• it is a logarithm with base e
• there is no y intercept
• the x intercept is at $\left(1,0\right)$
• the domain is $(0,\infty )$
• the range is $\left(-\infty ,\infty \right)$
• $\ln \left(e^x\right)=x$
• $e^{\ln \left(x\right)}=x$

Converting other logarithmic functions to natural logarithmic functions

It can be helpful to change the base of logarithmic functions to see how they compare to each other. To do this use the Proportion Rule for logarithms,

Since you want to convert ${\log }_b\left(x\right)$to ${\log }_e\left(x\right)$, use $a=e$ to get

${\log }_b\left(x\right)=\frac{{\log }_a\left(x\right)}{{\log }_a\left(b\right)}=\frac{{\log }_e\left(x\right)}{{\log }_e\left(e\right)}=\frac{\ln \left(x\right)}{\ln \left(b\right)}$

So $f\left(x\right)={\log }_b\left(x\right)$is equivalent to $f\left(x\right)=\frac{\ln \left(x\right)}{\ln \left(b\right)}$.

Derivatives of the Natural Logarithmic Function

The derivative of the natural logarithmic function is

$\frac{d}{dx}\ln \left(x\right)=\frac{1}{x}$

For more information on the derivative of the natural logarithmic function see Derivative of the Logarithmic Function.

Integration of Natural Logarithmic Functions

The integral of the natural logarithmic function is

$\int \ln \left(x\right)dx=x·\ln \left(x\right)-x+C$

For more information on the integral of the natural logarithmic function see Integrals Involving Logarithmic Functions.