Nielsen · Mathematics·Secondary Math 3

Why the log rules work

You can memorize the three log rules, or you can know where they come from. Each rule below has three explanations. Only one of them holds up. Your job is to find it.

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First, the three words

log₂ 8 = 3 means 2³ = 8

2 is the base
8 is the argument
3 is the exponent

A logarithm is an exponent. It answers: what exponent goes on the base to make the argument?
The argument is the number you are taking the log of.
Keep an eye on which number is playing which part. That is where the flawed explanations go wrong.

How this works

Read all three explanations before you choose.
Two of them have a flaw hidden in one of their steps. They still land on the right answer, so the answer cannot tell you which one to trust.
Check each step on its own. Is it actually true? Would it still be true with different numbers?
If you pick a flawed one, you get a test to run. Run it before you open the reveal.

The big idea

Every log rule is an exponent rule you already know, read from the exponent's point of view.

Exponent rule	Log rule	In words
b^x · b^y = b^{x + y}	log_b(M · N) = log_b M + log_b N	Multiply the arguments, add the exponents.
b^x ÷ b^y = b^{x − y}	log_b(M ÷ N) = log_b M − log_b N	Divide the arguments, subtract the exponents.
(b^x)^p = b^{p · x}	log_b(M^p) = p · log_b M	Raise the argument to a power, multiply the exponent by it.

Notice what every flawed explanation had in common: it got the right number on one example. A single example that works is not a reason. A real explanation says why, and keeps working when you change the numbers.