Using 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 \(\ln(−500)\).