Question

How do you solve `3^(2-x)=5^(2x+1)`?

Answer 1

I found `x=0.13614`

Explanation:

We can try taking the natural log of both sides:
`ln(3^(2-x))=ln(5^(2x+1))`
then use the property of logs:
`logx^y=ylogx` to write:
`(2-x)ln(3)=(2x+1)ln(5)`
`2ln(3)-xln(3)=2xln(5)+ln(5)`
collecting `x`:
`x(2ln(5)+ln(3))=2ln(3)-ln(5)`
so:
`x=(2ln(3)-ln(5))/(2ln(5)+ln(3))=0.13614`