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[Solved] Prove that log MN = log M + log N


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Prove that log MN = log M + log N


First all we are going to assume that all logarithms are to base 10.

  Then, let M = 10x and N = 10y

By the theory of logarithms,   M = 10x is the same as log10 M = x.

If you don't understand this part, then you can check this post about the theory of logarithms.

This also means N = 10y is the same as log10 N = y.

Now, moving on:

MN = (10x)(10y)

 According to the laws of indices:

MN = 10x+y

Then we take logarithms of both sides:

log MN = log 10x+y 

log MN = (x + y)log 10

Since the logs are to base 10, log 10 = 1

log MN = x + y

As we proved earlier, log M = x and log N = y.

Therefore log MN = log M + log N

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