Solve for x exactly: 8 ^2x = 4^5x-3

Answer 1

Hi Rebecca!

These are tricky questions, but we can figure them out. We are trying to solve:

8^2x = 4^5x-3

Let me start by looking at an easier problem:

8^x = 8²

This problem is much simpler since both of the base numbers are the same (they are both 8). When this happens, we just set the exponents equal: x = 2 and we are done! However, our problem is much harder because the bases are NOT the same -- one is an 8, the other is a 4. If only we could make them the same... well, we can!

Back to our actual problem:

8^2x = 4^5x-3

We know that 8 = 2³ and 4 = 2², so we can rewrite our problem as

(2³)^2x = (2²)^5x-3

Using the exponent multiplication rule, we have

2^6x = 2^2(5x-3)

Distributing gives us:

2^6x = 2^10x-6

Aha! Now that the bases are equal, we can see that 6x = 10x - 6 must be true. Solving this we get:

0 = 4x - 6

6 = 4x

6/4 = x

3/2 = x.

The answer is x = 3/2.

Hope that helps! Let me know if anything didn't make sense. :)