Question

How do you solve `2^ { 2x + 5} = ( \frac { 1} { 4} ) ^ { x - 1}`?

Answer 1

Change the bases to be equal to each other and solve for the exponents as if it's a linear equation.

Explanation:

2 can be written as an exponent with a base of `1/4`: `2=(1/4)^-2`

This means that we can rewrite the left side of the equation as `((1/4)^-2)^(2x+5)` or, using the rules of exponents, `(1/4)^(-4x-10)`

Now, going back to the original problem, it becomes apparent that `-4x-10=x-1` (again, using the rules of exponents)

This can be solved as a linear equation:

`-4x-10=x-1`
`-10=5x-1`
`-9=5x`
`-9/5=x`