How do you solve this?

4^m = 1/32

The key to solving this: recognizing powers of 32.
Since 4 = 2^2, and
32 = 2^5, then we have

(2^2)^m = 1/2^5

Whenever we have a power to a power, we can multiply the powers.

2^(2m) = 1/2^5

And whenever we have 1 over a positive exponent, we can make it a negative exponent after the conversion.

2^(2m) = 2^(-5)

Notice we have a base of 2 on both sides of the equation. Now, we can equate the exponents.

2m = -5
m = -5/2

2^(2m) = 2^(- 5)

2m = - 5

m = - 5/2

I would personally use natural logs to solve this. You can solve any problem like this if you know how how to use natural logs. So lets start

4^m=1/32
First I apply a natural log to both sides of the equation
ln(4^m)=ln(1/32)
Because of the properties of logarithms, I can take out that m on the left and have it multiply. Like this
m*ln(4)=ln(1/32)
Now from here the math turns to more basic algebra. This is your answer to m
m=ln(1/32)/ln(4)

That is the answer in its unsimplified form. However, if you look closely, this answer can be decomposed into something simpler. Again we need to use the properties of logarithms to decompose this to something simpler.

m=ln(1/32)/ln(4)
Below I use the properties of logs over and over again
m=(ln(1)-ln(32))/ln(4)
m=-ln(2^5)/ln(2^2)
m=(5*-ln(2))/(2*ln(2))
cancel the ln(2)
m=-5/2

I'm hoping you could follow that. If you couldn't then I highly recommend you check out pages like this to figure out how logs work-
http://uncw.edu/courses/mat111hb/EandL/logprop/logprop.html

Anyway. Sometimes you can simplify but other times you really can't. It all depends on what equation you're given to solve.

I hope this information helped you out.

You can also work with the natural logarithm with the knowledge that LN(x^y) = y*LN(x)

4^m = 1/32 | ln
m*LN(4) = LN(1/32)
m = LN(1/32)/LN(4) = -2.5

4^m = 1/32
(2^2)^m = 1/(2^5)
2^(2m) = 2^-5
2m = -5
m = -5/2

4^m=1/32=1/(2^5)=2^(-5)
2^2*m=2^(-5)
2m=-5
m=-5/2

I hope that you understand and good luck

4^m = 1/32

then 32(4^m) = 1

Now 32 = 2^5, and 4^m = (2^2)^m = 2^2m

Hence (2^5)(2^2m) = 1

i.e. 2^(5+2m) = 1

We can see that 5+2m = 0

Hence m = -2/5

4^m=1/32

(2^2)^m = 1/2^5

2^2m = 2^-5

Therefore 2m = -5
m = -5/2 = -2 1/2