Question

Simplify: $\log \left(\frac{70}{33}\right)+\log \left(\frac{22}{135}\right)-\log \left(\frac{7}{18}\right)$

Answer 1

Simplify the equation.

$$\begin{gathered} \log \left(\frac{70}{33}\right)+\log \left(\frac{22}{135}\right)-\log \left(\frac{7}{18}\right) \\ =\log \left(\frac{70}{33}\times \frac{22}{135}\right)-\log \left(\frac{7}{18}\right)\left[\because \log a+\log b=\log \left(ab\right),\log a-\log b=\log \left(\frac{a}{b}\right)\right] \\ =\log \left(\frac{70\times 22\times 18}{33\times 135\times 7}\right) \\ =\log \left(\frac{8}{9}\right) \\ =\log 8-\log 9 \\ =\log 2^3-\log 3^2 \\ =3\log \left(2\right)-2\log \left(3\right) \end{gathered}$$

Hence, $\log \left(\frac{70}{33}\right)+\log \left(\frac{22}{135}\right)-\log \left(\frac{7}{18}\right)$ can be written as $3\log \left(2\right)-2\log \left(3\right).$