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

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Answer 1 · 2018-05-29T01:46:23

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

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:

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

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