Question

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

Answer 1

Convert to logarithmic form.

Explanation:

`log2^(x + 1) = log3^x`

Simplify using the rule `loga^n = nloga`

`(x + 1)log2 = xlog3`

Distribute on the left side. Don't forget you cannot multiply directly a log with a non-log (e.g `2 xx log3!= log6`but is equal to `2log3`)

`xlog2 + log2 = xlog3`

Put the x's to one side of the equation.

`log2 = xlog3 - xlog2`

`log2 = x(log3 - log2)`

Simplify the right side of the equation further by using the rule `logm - logn = log(m / n)`

`log2 = x(log(3/2))`

`log2/log(3/2) = x`

You will want to ask your teacher if he/she wants the answer in exact form or rounded off. Just make sure to check.

Practice exercises:

1. Solve for x. Leave answers in exact form.

a) `2^(3x) = 5^(x + 1)`

b). `5^(x - 3) = 3^(2x + 1)`

Challenge problem

Find the value of x in `2^(4x - 6) = 5 xx 3^(x + 7)`