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The Laws of Logarithms

There are several “Laws” of Logarithms. They are:

The bases seem to cancel in the first expression and to swap with the argument in the second.

Ie we eliminate a log by raising the base to the power of both sides.

Problems involving logs usually reduce to a simple linear or quadratic equation.

Example:

Use the first rule to obtain a single log.

Use the sixth rule to eliminate the log.

Expand this and simplify.

Factorise and solve.

Example:

We must make the bases the same for both logs. We can do this with the second case of the fifth rule:We have,

Substitute to giveand multiply throughout by p:

Subtract and simplify to give,then factorise.

So,

or

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