Exponential equations with variables on both sides

Question

I have the following:

$$8^{3x+4} = 5^{4x-2}$$

How would I solve this? I tried this:

$$(3x+4)\log 8 = (4x-2)\log 5$$

but have no idea where to go from there. Thank you!

Answer 1

You've done the hard part. Here's the easy part: $$ 3x\log 8 + 4\log 8 = 4x\log 5- 2\log 5 $$ $$ 3x\log 8 - 4x\log 5 = -2\log 5 - 4\log 8 $$ $$ x(3\log8-4\log5) = -2\log 5 - 4\log 8 $$ $$ x = \frac{-2\log 5 - 4\log 8}{3\log8-4\log5} $$

Answer 2

$x\left(3\log(8)-4\log(5)\right)=-4\log(8)-2\log(5)$