## Use natural logarithms

The most frequently used base for logarithms is e. Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e logarithm, ${\mathrm{log}}_{e}\left(x\right)$, has its own notation, $\mathrm{ln}\left(x\right)$.

Most values of $\mathrm{ln}\left(x\right)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\mathrm{ln}1=0$. For other natural logarithms, we can use the $\mathrm{ln}$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

### A General Note: Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. We write ${\mathrm{log}}_{e}\left(x\right)$ simply as $\mathrm{ln}\left(x\right)$. The natural logarithm of a positive number x satisfies the following definition.

For $x>0$,

$y=\mathrm{ln}\left(x\right)\text{ is equivalent to }{e}^{y}=x$

We read $\mathrm{ln}\left(x\right)$ as, “the logarithm with base e of x” or “the natural logarithm of x.”

The logarithm y is the exponent to which e must be raised to get x.

Since the functions $y=e^{x}$ and $y=\mathrm{ln}\left(x\right)$ are inverse functions, $\mathrm{ln}\left({e}^{x}\right)=x$ for all x and $e^{\mathrm{ln}\left(x\right)}=x$ for $x>0$.

### How To: Given a natural logarithm with the form $y=\mathrm{ln}\left(x\right)$, evaluate it using a calculator.

1. Press [LN].
2. Enter the value given for x, followed by [ ) ].
3. Press [ENTER].

### Example 6: Evaluating a Natural Logarithm Using a Calculator

Evaluate $y=\mathrm{ln}\left(500\right)$ to four decimal places using a calculator.

### Solution

• Press [LN].
• Enter 500, followed by [ ) ].
• Press [ENTER].

Rounding to four decimal places, $\mathrm{ln}\left(500\right)\approx 6.2146$

### Try It 6

Evaluate $\mathrm{ln}\left(-500\right)$.

Solution