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, [latex]{\mathrm{log}}_{e}\left(x\right)\\[/latex], has its own notation, [latex]\mathrm{ln}\left(x\right)\\[/latex].

Most values of [latex]\mathrm{ln}\left(x\right)\\[/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\mathrm{ln}1=0\\[/latex]. For other natural logarithms, we can use the [latex]\mathrm{ln}\\[/latex] 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 [latex]{\mathrm{log}}_{e}\left(x\right)\\[/latex] simply as [latex]\mathrm{ln}\left(x\right)\\[/latex]. The natural logarithm of a positive number x satisfies the following definition.

For [latex]x>0\\[/latex],

[latex]y=\mathrm{ln}\left(x\right)\text{ is equivalent to }{e}^{y}=x\\[/latex]

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

How To: Given a natural logarithm with the form [latex]y=\mathrm{ln}\left(x\right)\\[/latex], 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 [latex]y=\mathrm{ln}\left(500\right)\\[/latex] to four decimal places using a calculator.

Solution

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

Rounding to four decimal places, [latex]\mathrm{ln}\left(500\right)\approx 6.2146\\[/latex]

Try It 6

Evaluate [latex]\mathrm{ln}\left(-500\right)\\[/latex].

Solution