Question

If `3^(x+2) = 2^(2x-1)` and `x = log A / log B`, what is the product `AB`?

Answer 1

`AB=1/24`

Explanation:

As `3^(x+2)=2^(2x-1)`, we have

`(x+2)log3=(2x-1)log2`

or `xlog3+2log3=2xlog2-log2`

or `x(log3-2log2)=-2log3-log2`

or `x(log(3/2^2))=log(3^(-2)xx2^(-1))`

or `xlog(3/4)=log(1/(3^2xx2))`

or `x=log(1/18)/log(3/4)`

Hence `A=1/18` and `B=3/4`

and `AB=1/18xx3/4=1/24`