Express the following as a single logarithm:

$\dfrac{1}{2}$log 25 - 2log 3 + 1

Given,

$\Rightarrow \dfrac{1}{2}\text{log 25 - 2log 3 + 1} \\[1em] \Rightarrow \dfrac{1}{2}\text{log 5}^2 - \text{2log 3 + log 10} \\[1em] \Rightarrow \dfrac{1}{2} \times 2\text{log 5} - \text{log 3}^2 + \text{log 10} \\[1em] \Rightarrow \text{log 5 - log 9 + log 10} \\[1em] \Rightarrow \text{log }\dfrac{5 \times 10}{9} \\[1em] \Rightarrow \text{log }\dfrac{50}{9}.$

Hence, $\dfrac{1}{2}\text{log 25 - 2log 3 + 1} = \text{log }\dfrac{50}{9}.$