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:

`(1)/(\log_4x)+(1)/(\log_5x)=(1)/(\log_(20)x)`

This is achieved thus:

We know that the reciprocal law of logarithm states:

`\log_ba=(1)/(\log_ab)`

Therefore, we have:

`\begin{gathered} From\text{ }the\text{ }left\text{ }hand\text{ }side, \\ (1)/(\log_4x)+(1)/(\log_5x)=\log_x4+\log_x5 \\ \\ \text{ Using the product law, we have} \\ =\log_x(4*5)=\log_x20 \\ \\ \text{ Using the reciprocal law, we have} \\ =\log_x20=(1)/(\log_(20)x) \\ \\ \text{ which is the right hand side } \end{gathered}`

Hence, we have proved that: