Question

How do you solve `2 times 3^x = 7 times 5^x`?

Answer 1

I have taken you up to the final calculation!

Explanation:

Given: `" "2xx3^x=7xx5^x`

Collecting like terms

`2/7=5^x/3^x`

Taking logs of both sides ( have chosen to use Natural logs)

`color(brown)("Note that when taking logs, division of the source values becomes subtraction of the logs.")`
`color(brown)("Also "log(b^x) " can be written as "xlog(b))`

`ln(2)-ln(7)=ln(5^x) - ln(3^x)`

This may be can be written as:

`ln(2)-ln(7)=xln(5) -x ln(3)`

`ln(2)-ln(7)=x(ln(5)-ln(3))`

`x=(ln(2)-ln(7))/(ln(5)-ln(3) )`

I will let you work this out!