Read: Evaluate Logarithms
Learning Objectives
- Mentally evaluate logarithms
- Define natural logarithm, evaluate natural logarithms with a calculator
- Define common logarithm, evaluate common logarithms mentally and with a calculator
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’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].
Example
Solve [latex]y={\mathrm{log}}_{4}\left(64\right)[/latex] without using a calculator.Answer:
First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[/latex]. Next, we ask, "To what exponent must [latex]4[/latex] be raised in order to get [latex]64[/latex]?"
We knowTherefore,
Example
Evaluate [latex]y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)[/latex] without using a calculator.Answer:
First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\frac{1}{27}[/latex]. Next, we ask, "To what exponent must [latex]3[/latex] 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
Therefore, [latex]{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3[/latex].
How To: Given a logarithm of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex], evaluate it mentally.
- Rewrite the argument x as a power of b: [latex]{b}^{y}=x[/latex].
- Use previous knowledge of powers of b 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],
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].
Example
Evaluate [latex]y=\mathrm{ln}\left(500\right)[/latex] to four decimal places using a calculator.Answer:
- Press [LN].
- Enter [latex]500[/latex], followed by [ ) ].
- Press [ENTER].
Rounding to four decimal places, [latex]\mathrm{ln}\left(500\right)\approx 6.2146[/latex]
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] simpliy 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] is equivalent to [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.Answer: We know [latex]10^3=1000[/latex], therefore [latex-display]{\mathrm{log}}_{}(1000)=3[/latex-display]
Example
Evaluate [latex]y={\mathrm{log}}_{}(321)[/latex] to four decimal places using a calculator.Answer:
- Press [LOG].
- Enter [latex]321[/latex], followed by [ ) ].
- Press [ENTER].
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?Answer: We begin by rewriting the exponential equation in logarithmic form.
[latex]10^x=500[/latex]
[latex]{\mathrm{log}}_{}(500)=x[/latex]
Next we evaluate the logarithm using a calculator:
- Press [LOG].
- Enter [latex]500[/latex] followed by [ ) ].
- Press [ENTER].
- To the nearest thousandth, [latex]{\mathrm{log}}_{}(500)\approx2.699[/latex]
log ( 500 ) ≈ 2.699. log ( 500 ) ≈ 2.699.
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 sue base 10, and often use base e. Logarithms can also be evaluated with most kinds of calculator.Licenses & Attributions
CC licensed content, Shared previously
- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
- Ex 1: Evaluate Logarithms Without a Calculator - Whole Numbers. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Ex: Evaluate Natural Logarithms on the Calculator. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
