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I noticed that order of operations matters when applying logarithm rules.

I'll use a different example from my homework problem to illustrate what I'm talking about.

ln(5*2^3) does not equal 3*(ln(5*2))

Apparently you have to do ln5+3*ln2 - you have to separate out the products before multiplying the second term by the exponent! It kind of makes sense but I can't say exactly why this is how it has to be.

Can anyone confirm that what I have said is true and explain why?