Learning Objectives
- 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.
- 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?”
Example: Solving Logarithms Mentally
Solve [latex]y={\mathrm{log}}_{4}\left(64\right)[/latex] without using a calculator.
Try It
Solve [latex]y={\mathrm{log}}_{121}\left(11\right)[/latex] without using a calculator.
Example: Evaluating the Logarithm of a Reciprocal
Evaluate [latex]y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)[/latex] without using a calculator.
Try It
Evaluate [latex]y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)[/latex] without using a calculator.
Use common logarithms
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].
Example: Converting from Exponential Form to Logarithmic Form
Write the following exponential equations in logarithmic form.
- [latex]{2}^{3}=8[/latex]
- [latex]{5}^{2}=25[/latex]
- [latex]{10}^{-4}=\frac{1}{10,000}[/latex]
Try It
Write the following exponential equations in logarithmic form.
- [latex]{3}^{2}=9[/latex]
- [latex]{5}^{3}=125[/latex]
- [latex]{2}^{-1}=\frac{1}{2}[/latex]
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, [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 equivalent 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 with the form [latex]y=\mathrm{ln}\left(x\right)[/latex], evaluate it using a calculator.
- Press [LN].
- Enter the value given for x, followed by [ ) ].
- 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.
Try It
Evaluate [latex]\mathrm{ln}\left(-500\right)[/latex].