Question

without using e or in, prove (image attached greatly appreciated)

Answer 1

Explanation

We are required to use logarithm properties/laws to prove the given equation below:

$\frac{1}{\log_4x}+\frac{1}{\log_5x}=\frac{1}{\log_{20}x}$

This is achieved thus:

We know that the reciprocal law of logarithm states:

$\log_ba=\frac{1}{\log_ab}$

Therefore, we have:

$\begin{gathered} From\text{ }the\text{ }left\text{ }hand\text{ }side, \\ \frac{1}{\log_4x}+\frac{1}{\log_5x}=\log_x4+\log_x5 \\ \\ \text{ Using the product law, we have} \\ =\log_x(4\times5)=\log_x20 \\ \\ \text{ Using the reciprocal law, we have} \\ =\log_x20=\frac{1}{\log_{20}x} \\ \\ \text{ which is the right hand side } \end{gathered}$

Hence, we have proved that:

$\frac{1}{\operatorname{\log}_{4}x}+\frac{1}{\operatorname{\log}_{5}x}=\frac{1}{\operatorname{\log}_{20}x}$