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

Now consider solving ${\mathrm{log}}_{7}49$ and ${\mathrm{log}}_{3}27$ mentally.

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

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate ${\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}$ mentally.

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

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, ${\mathrm{log}}_{e}\left(x\right)$, has its own notation, $\mathrm{ln}\left(x\right)$.

Most values of $\mathrm{ln}\left(x\right)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\mathrm{ln}1=0$. For other natural logarithms, we can use the $\mathrm{ln}$ 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.

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