Question

Hey, everyone, I was just looking for some help with solving this question (3^x)*27=(9^x-4) apparently, it is looking for an answer set?

Answer 1

`x = 2.995" or "-1.733`

But : see below

Explanation:

As it is written, the equation seems to be

`color(red)((3^x)times 27=(9^x-4))`

Let us denote `3^x` by `y`. Then `9^x = (3^2)^x = (3^x)^2 = u^2`
So, the equation becomes

`27u = u^2-4 qquad implies qquad u^2-27u+4 = 0`

Thus

`u = (27 pm sqrt(27^2-4times1times 4))/2`

Thus

`3^x = u = 26.85` or `3^x =0.1500`

So

`x = ln(3^x)/ln3 = 2.995" or "-1.733`

But maybe the equation really was

`color(blue)((3^x)times 27=9^(x-4))`

Then,

`3^x times 3^3 = (3^2)^{x-4}`

and so

`3^(x+3) = 3^(2x-8)`

which means that

`x+3 = 2x-8 qquad implies qquad color(blue)(x = 11)`