Question

Express each of the following as a single logarithm: $\frac{1}{2}\log \left(36\right)+2\log \left(8\right)-\log \left(1.5\right)$.

Answer 1

Simplify the expression.

Given expression is $\frac{1}{2}\log \left(36\right)+2\log \left(8\right)-\log \left(1.5\right)$

$$\begin{gathered} \therefore \frac{1}{2}\log \left(36\right)+2\log \left(8\right)-\log \left(1.5\right) \\ =\frac{1}{2}\log \left(6^2\right)+2\log \left(2^3\right)-\log \left(\frac{15}{10}\right) \\ =\log \left(6\right)+6\log \left(2\right)-\log \left(\frac{15}{10}\right)\left[\because log\left(a^n\right)=nlog\left(a\right)\right] \\ =\log \left(2\times 3\right)+6\log \left(2\right)-\log \left(\frac{3}{2}\right) \\ =\log \left(2\right)+\log \left(3\right)+6\log \left(2\right)-\log \left(3\right)+\log \left(2\right)\left[\because log\left(ab\right)=log\left(a\right)+log\left(b\right),log\left(\frac{a}{b}\right)=log\left(a\right)-log\left(b\right)\right] \\ =8\log \left(2\right) \end{gathered}$$

Hence, the required single logarithm is $8\log \left(2\right)$.