Express the following as a single logarithm:

$\dfrac{1}{2}$log 36 + 2log 8 - log 1.5

Given,

$\Rightarrow \dfrac{1}{2}\text{log 36 + 2log 8 - log 1.5} \\[1em] \Rightarrow \dfrac{1}{2}\text{log 6}^2 + \text{log 8}^2 - \text{log} \dfrac{15}{10} \\[1em] \Rightarrow \dfrac{1}{2} \times 2 \times \text{log 6 + log 64 - (log 15 - log 10)} \\[1em] \Rightarrow \text{log 6 + log 64 - log 15 + log 10} \\[1em] \Rightarrow \text{log} \dfrac{6 \times 64 \times 10}{15} \\[1em] \Rightarrow \text{log 256}.$

Hence, $\dfrac{1}{2}\text{log 36 + 2log 8 - log 1.5}$ = log 256.