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Can someone help me with log laws?

We're starting to learn about logarithms at school and our teacher introduced us to the log laws. I understand them but was wondering how the following log laws work/can be proved. I asked my teacher but she wasn't sure and I was wondering if someone knew how the following can equate.

ylogax = logax^y
a^logam = m
logab = (logcb)/(logca) --> (change of base)

sorry, can't do subscript for the bases and superscript for powers, hopefully it's understandable though

I'm not going to give formal proofs. I'll just give an example that might help you see how they operate.

log(base a)x is the power to which you have to raise a in order to get x. That's the definition of a logarithm, and that definition is what your second rule is stating.

Let's pick a=2, and x=16.

What power do we have to raise 2 to in order to get to 16? 4, of course. 2*2*2*2=16. This can be written as 2^4=16. This also means that log(base 2)16 = 4.

Now, let's see what happens if we cube that 16. 16*16*16=4096.

log(base 2)4096 = 12 (check this for yourself if you want)

We can also write this as:
log(base 2)(16^3) = 12

Remember that log(base 2)16 = 4.
Notice that 3*4=12

This same thing works out for any other values of a, x, and y. (except, of course, weird values of a such as a=0 and a=1)

Now, for your third rule.
log(base 16)4096 = log(base 2)4096/log(base 2)16
3 = 12/4

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