Evaluate :

$\dfrac{\text{log}$5 \space 8}{\text{log }{25} \space 16 \times \text{log }_{100} \space 10}

Simplifying the expression,

$\Rightarrow \dfrac{\text{log}$5 \space 8}{\text{log }{25} \space 16 \times \text{log }{100} \space 10} \\[1em] \Rightarrow \dfrac{\text{log}5 \space 8}{\text{log }{5^2} \space 16 \times \text{log}{10^2} \space 10} \\[1em] \Rightarrow \dfrac{\text{log}5 \space 8}{\dfrac{1}{2} \times \text{log }{5} \space 16 \times \dfrac{1}{2} \times \text{log}{10} \space 10} \\[1em] \Rightarrow \dfrac{\text{log}5 \space 8}{\dfrac{1}{4} \times \text{log}{5} \space 16} \\[1em] \Rightarrow \dfrac{4\text{log}5 \space 8}{\text{log}_{5} \space 16} \\[1em] \Rightarrow \dfrac{4 \times \dfrac{\text{log 8}}{\text{log 5}}}{\dfrac{\text{log 16}}{\text{log 5}}} \\[1em] \Rightarrow \dfrac{\text{4 log 8. log 5}}{\text{log 16. log 5}} \\[1em] \Rightarrow \dfrac{\text{4 log 8}}{\text{log 16}} \\[1em] \Rightarrow \dfrac{\text{4 log 2}^3}{\text{log 2}^4} \\[1em] \Rightarrow \dfrac{4 \times 3 \times \text{log 2}}{4 \times \text{log 2}} \\[1em] \Rightarrow \dfrac{12}{4} \\[1em] \Rightarrow 3.

Hence, $\dfrac{\text{log}$5 \space 8}{\text{log }{25} \space 16 \times \text{log }_{100} \space 10} = 3.