Question
Evaluate: $\frac{\log _5 8}{\log _{25} 16 \times \log _{100} 10}$
✓
Answer
$ \frac{\log _5 8}{\log _{25} 16 \times \log _{100} 10}$
$ =\frac{\frac{\log _{10} 8}{\log _{10} 5}}{\frac{\log _{10} 16}{\log _{10} 25} \times \frac{\log _{10} 10}{\log _{10} 100}}$
$ =\frac{\frac{\log _{10} 2^3}{\log _{10} 5}}{\frac{\log _{10} 2^4}{\log _{10} 5^2} \times \frac{\log _{10} 10}{\log _{10} 10^2}}$
$ =\frac{\log _{10} 2^3}{\log _{10} 5} \times \frac{\log _{10} 5^2}{\log _{10} 2^4} \times \frac{\log _{10} 10^2}{\log _{10} 10}$
$ =\frac{3 \log _{10} 2}{\log _{10} 5} \times \frac{2 \log _{10} 5}{4 \log _{10} 2} \times \frac{2 \log _{10} 10}{\log _{10} 10}$
$ =3$
$ =\frac{\frac{\log _{10} 8}{\log _{10} 5}}{\frac{\log _{10} 16}{\log _{10} 25} \times \frac{\log _{10} 10}{\log _{10} 100}}$
$ =\frac{\frac{\log _{10} 2^3}{\log _{10} 5}}{\frac{\log _{10} 2^4}{\log _{10} 5^2} \times \frac{\log _{10} 10}{\log _{10} 10^2}}$
$ =\frac{\log _{10} 2^3}{\log _{10} 5} \times \frac{\log _{10} 5^2}{\log _{10} 2^4} \times \frac{\log _{10} 10^2}{\log _{10} 10}$
$ =\frac{3 \log _{10} 2}{\log _{10} 5} \times \frac{2 \log _{10} 5}{4 \log _{10} 2} \times \frac{2 \log _{10} 10}{\log _{10} 10}$
$ =3$
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