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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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lionel operating log dumpcar 6 19473
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LIONEL POSTWAR 3364 OPERATING DUMP CAR WITH 3 LOGS 1965 66 1968
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LIONEL POSTWAR 3361 OPERATING LOG DUMP CAR AS IS NO RETURN 1955 1958
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Lionel 3461x Operating Dump Car With No 160 Bin And logs
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LIONEL 3461 GREEN OPERATING LOG DUMP CAR W OE BIN  LOGS READY TO USE LOT 28
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USED VINTAGE LIONEL O GAUGE 3461 OPERATING LOG DUMP CAR WITH 5 LOGS
USED VINTAGE LIONEL O GAUGE 3461 OPERATING LOG DUMP CAR WITH 5 LOGS
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NICE USED LIONEL 24152 OPERATING 364 LOG LOADER WITH ORIGINAL BOX
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Marx Searchlight Flat Car And Operating Log Dump Car For 027 Gauge
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AMERICAN FLYER 787 MTH 30 9119 OPERATING LOG LOADER MINT S O GAUGE C 10
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Vintage Lionel Trains O O 27 Operating Union Pacific Log Dump Car 6 9303 NOS
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LIONEL 6 9313 AtSF OPERATING LOG DUMP CAR In Box VG condition
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Lionel Operating Log Dump Car 6 9303 P4
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LIONEL MODERN 9303 OPERATING LOG DUMP CAR NEW IN ORIG BOX 1974 1978
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LIONEL POSTWAR 3361 OPERATING LOG DUMP CAR 1955 1958 EXCELLENT CONDITION
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LIONEL 74 Operating Log Dump Car UP 6 9303 Original Box
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