Algebra help? How do you rewrite logarithms in exponential form

For the problem:

log (2x) = y how is it written in exponential form?

Also, how do you solve for x in the problem:

4^x = 32^2x-1

2x=10^y the base of log is 10. I remember it by saying the base becomes the base.

For the next problem you need to make both sides have 2 to some power. It looks like this:

4^x = 32^2x-1
2^2x = 2^5(2x-1)
2^2x = 2^(10x-5)

Since both bases are 2 then the exponents must be equal to each other.

2x = 10x - 5

You should be able to solve that equation from there.

Assuming the log means log base 10, it would be written as 2x=10^y
For the 2nd problem keep in mind 32 is 4^2.5 so it can be rewritten as 4^x = 4^(5x-2.5).
Set both of these to log base 4 so it basically is x = 5x-2.5
Then you can combine like terms, eventually getting -4x = -2.5, then multiply by -1 to get
4X = 2.5 divide by 4 and X=.625

log₁₀(2x) = y
2x = 10^y
:::::
4^x = 32^(2x - 1)
(2²)^x = (2⁵)^(2x - 1)
2^(2x) = 2^(10x-5)
2x = 10x - 5
8x = 5
x = 0.625