Question

How do you solve `logx^2 + logx^3 / log(100x) = 3`?

Answer 1

The solutions are `x=10^{-3}=1/1000=0.001` and `x=10`.

Explanation:

First, use properties of logarithms to rewrite the equation `log(x^2)+(log(x^3))/(log(100x))=3` as `2log(x)+(3log(x))/(log(100)+log(x))=3`, or

`2log(x)+(3log(x))/(2+log(x))=3`

Next, you can multiply everything by `2+log(x)` to get `4log(x)+2(log(x))^2+3log(x)=6+3log(x)`

Eliminate `3log(x)` since it is found on both sides.

`4log(x)+2(log(x))^2=6`

Divide all terms by `2`.

`2log(x)+(log(x))^2=3`

Subtract `3` from both sides.

`2log(x)+(log(x))^2-3=0`

Rewrite.

`(log(x))^2+2log(x)-3=0`

Factor.

`(log(x)+3)(log(x)-1)=0`.

Therefore, we seek values of `x` so that `log(x)=-3` and `log(x)=1`, giving `x=10^{-3}=1/1000=0.001` and `x=10`. You can check that these both work in the original equation.