Question

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

Answer 1

You must first convert to logarithmic form.

Explanation:

`log5^(x - 1) = log3^x`

`(x - 1)log5 = xlog3`

Distribute the parentheses:

`xlog5 - log5 = xlog3`

Put all x terms to the left side of the equation:

`xlog5 - xlog3 = log5`

Factor out the x:

`x(log5 - log3) = log5`

Use the quotient rule:

`x(log(5/3)) = log5`

`x = log5/(log(5/3)`

`x = log_(5/3)5`

Practice exercises:

1. Solve for x. Leave in logarithmic form.

a) `2^(x - 2) = 3^(2x + 4)`

b) `3^(3x) = 4 xx 5^(x - 6)`

Good luck!