Question

How do you simplify `Log(x+4)=2log(x-2)`?

Answer 1

`x=5`

Explanation:

Simplify the right hand side using the logarithm rule:

`color(white)(sss)alog_b(c)=log_b(a^c)`

`log(x+4)=log((x-2)^2)`

Exponentiate both sides, which undoes both logarithms, leaving us with:

`x+4=(x-2)^2`

Distribute and simplify.

`x+4=x^2-4x+4`

`0=x^2-5x`

`0=x(x-5)`

`x=0,5`

However, the answer `x=0` is thrown out, since plugging in `0` in `2log(x-2)` would result in having to find the logarithm of a negative number, which is impossible.

Thus, the only valid answer is

`x=5`

graph{log(x+4)-2log(x-2) [-8.24, 17.07, -3.01, 9.65]}