Question

How to do this?


`{3}^(2 - x) = {5}^(x)`
the ans is

`( ln(9) )/( ln(15) )`

Answer 1

This equation uses two properties of logarithms:


`\ln(g^x) = x \ln(g)`


`\ln(a) + \ln(b) = \ln(ab)`

So you could take the ln from left and right hand side in the equation, and get:

(2-x)ln 3 = x ln 5

then

2 ln 3 - x ln 3 - x ln 5 = 0 =>

x(ln 3 + ln 5) = 2 ln 3

so x = 2 ln 3 / (ln3 + ln5)

Now using the 1st property you can say 2 ln 3 is ln 3² = ln9
and using the 2nd property you can say ln3 + ln5 = ln15

so x= ln9 / ln15