Question

Please can someone help me solve this:

Log ^9/4 base 2/3


Pls I need a well detailed explanation

Answer 1

We're given,

$\log_{\frac{2}{3}}\left(\dfrac{9}{4}\right)\quad\implies\quad\dfrac{\log\left(\dfrac{9}{4}\right)}{\log\left(\dfrac{2}{3}\right)}$

It's simpler to get to know that,

$\left(\dfrac{2}{3}\right)^{-2}\ =\ \left(\dfrac{3}{2}\right)^2\ =\ \dfrac{9}{4}$

So we can write,

$\dfrac{\log\left(\dfrac{9}{4}\right)}{\log\left(\dfrac{2}{3}\right)}\ =\ \dfrac{\log\left(\dfrac{2}{3}\right)^{-2}}{\log\left(\dfrac{2}{3}\right)}\ =\ \dfrac{-2\log\left(\dfrac{2}{3}\right)}{\log\left(\dfrac{2}{3}\right)}\ =\ \mathbf{-2}$

Hence -2 is the answer.