# How do you solve 9^(2x)=3^(2x+4) using a graph?

Oct 12, 2017

$x = 2$

#### Explanation:

We seek a solution of:

${9}^{2 x} = {3}^{2 x + 4}$

If we plot both functions on the same graph we have:
graph{ (y-9^(2x)) (y-3^(2x+4)) =0 [-5, 5, -2, 40]}

It appears as if there is no solution, except asymptotically as $x \rightarrow - \infty$

We can examine this analytically, as:

${9}^{2 x} = {3}^{2 \left(x + 2\right)}$
$\therefore {9}^{2 x} = {9}^{x + 2}$
$\therefore 2 x = x + 2$
$\therefore x = 2$

And so we conclude there is indeed a solution, but due to scale we are missing the solution, if we change the scale we get:
graph{ (y-9^(2x)) (y-3^(2x+4)) =0 [-5, 5, -500, 9000]}