4.3 Logarithmic functions  (Page 3/9)

Evaluating logarithms

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

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

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

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

Using common logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression $\log \left(x\right)$ means ${\log }_{10}\left(x\right).$ We call a base-10 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.