The Rules for Logarithms

In less formal terms, these three log rules might be expressed as:

Warning: Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "log_d(m) + log_b(n)" cannot be simplified, because the bases (the d and the b) are not the same, just as x² × y³ cannot be simplified because the bases (the x and y) are not the same.

Below are some examples of these log rules at work, using the base-10 common log:

It's all good and well to state these relationships. But how do we know that these rule are true? (If you don't care — which is fine — continue to the rest of the rules.)

Proofs of Product, Quotient, and Power Rules

I'll start with proving the Power Rule first, because I'll be using the Power Rule to prove the Product and Quotient Rules. (In the following proofs, x and y are placeholders with unspecified values, that I'm using to make my life easier.)

This proves the Power Rule.

With the Power Rule proven, I can now prove the Product and Quotient Rules.

This proves the Product Rule. The same sort of reasoning will prove the Quotient Rule.

This proves the Quotient Rule.

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How do you use the rules for logs?

In your algebra class, you'll use the log rules to "expand" and "condense" logarithmic expressions. The expanding is what I did in the first in each pair of examples above; the condensing is the second in each pair.

However, in other classes (such as calculus), you will need these rules to simplify complicated log expressions andconvert painful log expressions into nicer forms that will allow other mathematics to be done.

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There are additional log rules:

Note that, in all cases, the logarithm's base b must be positive and not equal to 1, and all values inside logarithms must be positive. (If you want to go straight to using these rules, skipping their proofs, then continue on to the next page.)

Rules (4) and (5) above are based on the fact that anything raised to the power 1 is just itself, and anything raised to the power 0 is just 1:

Rules (6) and (7) are based on the definition of logs as being themselves exponents. (You don't actually "cancel" anything, but it kinda feels like you do, hence my names for these rules.)

The Change-of-Base Formula can (and will) be incredibly helpful to you when you're working with logs on your calculator. You may be asked to evaluate a log expression where the log's base is something other than 10 or e. But your calculator can evaluate *only* logs with one of these two bases. The Change-of-Base Formula gives you a way out: you can convert the log expression into an equivalent expression that uses one of the two bases that your calculator is able to handle.

Proof of the Change-of-Base Formula:

To prove the Change-of-Base Formula, I'll use the same techniques of renaming as I used in the proofs above. The steps will be a bit more involved, though.

This proves the Change-of-Base Formula.

The only remaining rule is the Base-Switch Rule.

Note: Before I'd checked online for all the rules for logs (to make sure I wasn't forgetting anything), I'd never even heard of the Base-Switch Rule. I don't know where you'd use it.