We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > MATH 1314: College Algebra

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

Most values of ln(x)\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, ln1=0\mathrm{ln}1=0\\. For other natural logarithms, we can use the ln\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 loge(x){\mathrm{log}}_{e}\left(x\right)\\ simply as ln(x)\mathrm{ln}\left(x\right)\\. The natural logarithm of a positive number x satisfies the following definition.

For x>0x>0\\,

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

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

How To: Given a natural logarithm with the form y=ln(x)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=ln(500)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, ln(500)6.2146\mathrm{ln}\left(500\right)\approx 6.2146\\

Try It 6

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

Solution

Licenses & Attributions