Evaluating Logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, “base to the exponent gives us the number.”
[latex]\\[/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].
- 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].
- Rewrite the argument [latex]x[/latex] as a power of [latex]b[/latex]: [latex]{b}^{y}=x[/latex].
- Use previous knowledge of powers of [latex]b[/latex] to identify [latex]y[/latex] by asking, “To what exponent should [latex]b[/latex] be raised in order to get [latex]x[/latex]?”