Common and 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, $log_e(x)$, has its own notation, $\ln(x)$.

Most values of $\ln(x)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\ln1=0$. For other natural logarithms, we can use the $\ln$ 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.

DEFINITION OF THE NATURAL LOGARITHM

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

For $x > 0$,

$y=\ln (x) \quad$ is equivalent to $e^{y}=x$

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

HOW TO

Given a natural logarithm with the form $y= \ln(x)$, evaluate it using a calculator.

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

EXAMPLE 8

Evaluating a Natural Logarithm Using a Calculator

Evaluate $y= \ln(500)$ to four decimal places using a calculator.

Solution

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

Rounding to four decimal places, $\ln(500) \approx 6.2146$

TRY IT #8

Evaluate $\ln(−500)$.