Evaluating Logarithms

Learning Outcome

  • Evaluate logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\mathrm{log}}_{2}8[/latex]. We ask, “To what exponent must [latex]2[/latex] be raised in order to get [latex]8[/latex]?” Because we already know [latex]{2}^{3}=8[/latex], it follows that [latex]{\mathrm{log}}_{2}8=3[/latex].

Now consider solving [latex]{\mathrm{log}}_{7}49[/latex] and [latex]{\mathrm{log}}_{3}27[/latex] mentally.

  • We ask, “To what exponent must [latex]7[/latex] be raised in order to get [latex]49[/latex]?” We know [latex]{7}^{2}=49[/latex]. Therefore, [latex]{\mathrm{log}}_{7}49=2[/latex]
  • We ask, “To what exponent must [latex]3[/latex] be raised in order to get [latex]27[/latex]?” We know [latex]{3}^{3}=27[/latex]. Therefore, [latex]{\mathrm{log}}_{3}27=3[/latex]

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let us evaluate [latex]{\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}[/latex] mentally.

  • We ask, “To what exponent must [latex]\frac{2}{3}[/latex] be raised in order to get [latex]\frac{4}{9}[/latex]? ” We know [latex]{2}^{2}=4[/latex] and [latex]{3}^{2}=9[/latex], so [latex]{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}[/latex]. Therefore, [latex]{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2[/latex].

In our first example, we will evaluate logarithms mentally.

Example

Solve [latex]y={\mathrm{log}}_{4}\left(64\right)[/latex] without using a calculator.

In our first video, we will show more examples of evaluating logarithms mentally; this helps you get familiar with what a logarithm represents.

In our next example, we will evaluate the logarithm of a reciprocal.

Example

Evaluate [latex]y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)[/latex] without using a calculator.

 

How To: Given a logarithm of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex], evaluate it mentally

  1. Rewrite the argument x as a power of b: [latex]{b}^{y}=x[/latex].
  2. Use previous knowledge of powers of b to identify y by asking, “To what exponent should b be raised in order to get x?”

 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 [latex]1[/latex] is always [latex]0[/latex] 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{ can be written as }{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 x > [latex]0[/latex].

In the next example, we will evaluate a natural logarithm using a calculator.

Example

Evaluate [latex]y=\mathrm{ln}\left(500\right)[/latex] to four decimal places using a calculator.

In our next video, we show more examples of how to evaluate natural logarithms using a calculator.

Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is [latex]10[/latex]. In other words, the expression [latex]{\mathrm{log}}_{}[/latex] means [latex]{\mathrm{log}}_{10}[/latex]. We call a base-[latex]10[/latex] logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of Common Logarithm: Log is an exponent

A common logarithm is a logarithm with base [latex]10[/latex]. We write [latex]{\mathrm{log}}_{10}(x)[/latex]  simply as [latex]{\mathrm{log}}_{}(x)[/latex].  The common logarithm of a positive number, x, satisfies the following definition:

For [latex]x\gt0[/latex],

[latex]y={\mathrm{log}}_{}(x)[/latex] can be written as [latex]10^y=x[/latex]

We read [latex]{\mathrm{log}}_{}(x)[/latex] as ” the logarithm with base [latex]10[/latex] of x” or “log base [latex]10[/latex] of x”.

The logarithm y is the exponent to which 10 must be raised to get x.

Example

Evaluate [latex]{\mathrm{log}}_{}(1000)[/latex] without using a calculator.

Example

Evaluate [latex]y={\mathrm{log}}_{}(321)[/latex] to four decimal places using a calculator.

In our last example, we will use a logarithm to find the difference in magnitude of two different earthquakes.

Example

The amount of energy released from one earthquake was [latex]500[/latex] times greater than the amount of energy released from another. The equation [latex]10^x=500[/latex] represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Summary

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally because the logarithm is an exponent. Logarithms most commonly use base 10 and natural logarithms use base e. Logarithms can also be evaluated with most kinds of calculator.