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Study Guides > College Algebra

Use common 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 > 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 5: 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 5

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

Solution

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