Education Maths help - index notation

Im stuck on a question and im hoping you can help me out.
The question is asking me to solve:
4^{2x} times 8^{x-1} = 32
but i have no idea where to start as my notes don't have anything like this and i cant find any examples in the textbook. Im thinking maybe i have to change all the bases so that they are equal.

 

Long time since I did this, but I think you would need to get the first numbers the same, finding the common factor:
2^2 = 4 so 4^{2x} = 2^{2x+2}
2^3 = 8 so 8^{x-1} = 2^{x-1 +2} =2^{x+2}

Then for multiplication I think you add the powers together:
2^{2x} times 2^{x+2} = 2^{3x+2} = 32

Then you work out x from 2^{3x+2} = 32


Right?

 

Take logs of both sides...

 

That's what I'd do tbh, but then again it sounds like there's a non-log way.

[edit]

But by that eqn X=1, since 2^5 = 32. But if you sub that into the original equation you get 4^2 * 8^1 which != 32...

 

Good point, I should have tried to take it all the way through.

 

Classic...

And yes, the other method is reduce the base integers (which are 2, 8, and 32) down to 2.

So 8 will become:
2^(3)

But according to the rules of exponents, you need to add the new power to the existing (x-1) to make from 8:
2^(3+x-1)

The same steps for reducing 32 to a power of 2, which is:
2^(5)

Then you drop the mundane bases of 2 (make them disappear) and work only with the exponents as if they were whole bases in their own right while keep the operands that were between the original bases.

So you will get:
(2x+2) * (3+x-1) = 5

and work from there.

 

Oh I see - sad that I couldn't see the whole base-change idea... I suppose logs is a better way to fanny around with it, and that's all you've done there really...