Evaluating Logarithms | College Algebra - Lumen Learning

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Module 11: Exponential and Logarithmic Functions

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Evaluating Logarithms

Learning Outcomes

• Evaluate logarithms with and without a calculator.
• Evaluate logarithms with base 10 and base e.

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 2 be raised in order to get 8?” 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 7 be raised in order to get 49?” We know [latex]{7}^{2}=49[/latex]. Therefore, [latex]{\mathrm{log}}_{7}49=2[/latex].
• We ask, “To what exponent must 3 be raised in order to get 27?” 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’s 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].

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?”

Example: Solving Logarithms Mentally

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

Show Solution

First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[/latex]. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know [latex]{4}^{3}=64[/latex]

Therefore,

[latex]\mathrm{log}{}_{4}\left(64\right)=3[/latex]

Try It

Solve [latex]y={\mathrm{log}}_{121}\left(11\right)[/latex] without using a calculator.

Show Solution

[latex]{\mathrm{log}}_{121}\left(11\right)=\frac{1}{2}[/latex] (recall that [latex]\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11[/latex] )

Example: Evaluating the Logarithm of a Reciprocal

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

Show Solution

First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\frac{1}{27}[/latex]. Next, we ask, “To what exponent must 3 be raised in order to get [latex]\frac{1}{27}[/latex]“?

We know [latex]{3}^{3}=27[/latex], but what must we do to get the reciprocal, [latex]\frac{1}{27}[/latex]? Recall from working with exponents that [latex]{b}^{-a}=\frac{1}{{b}^{a}}[/latex]. We use this information to write

[latex]\begin{array}{l}{3}^{-3}=\frac{1}{{3}^{3}}=\frac{1}{27}\hfill \end{array}[/latex]

Therefore, [latex]{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3[/latex].

Try It

Evaluate [latex]y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)[/latex] without using a calculator.

Show Solution

[latex]{\mathrm{log}}_{2}\left(\frac{1}{32}\right)=-5[/latex]

Using 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 equal 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 Of 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: Evaluating a Natural Logarithm Using a Calculator

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

Show 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

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

Show Solution

It is not possible to take the logarithm of a negative number in the set of real numbers.

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• Question ID 35022. Authored by: Smart,Jim, mb Sousa,James. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
• College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/[email protected]. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/[email protected]

Licenses and Attributions CC licensed content, Original

• Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution

CC licensed content, Shared previously

• Question ID 35022. Authored by: Smart,Jim, mb Sousa,James. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
• College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/[email protected]. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/[email protected]

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