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, loge(x)loge(x), has its own notation, ln(x)ln(x).

Most values of ln(x)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=0ln1=0. For other natural logarithms, we can use the lnln 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 loge(x)loge(x) simply as ln(x)ln(x). The natural logarithm of a positive number x satisfies the following definition.

For x>0x>0,

y=ln(x) is equivalent to ey=xy=ln(x) is equivalent to ey=x

We read ln(x)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=exy=ex and y=ln(x)y=ln(x) are inverse functions, ln(ex)=xln(ex)=x for all x and eln(x)=xeln(x)=x for x>0x>0.

How To: Given a natural logarithm with the form y=ln(x)y=ln(x), 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=ln(500)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)≈6.2146ln(500)≈6.2146

Try It 6

Evaluate ln(−500)ln(−500).

Solution