## Rules of Logarithms

### Calculating Log values

Lets look at an example log.

This tells us that

The general form is

Examples

Calculate the following

We need to find out what power we raise 5 to that gives the answer 125.

Since it is clear that

We need to find out what power we raise 3 to that gives the answerSo

We need to find out what power we raise 10 to that gives the answerSo

Anything to the power of zero equals 1.So it follows that any log of 1 0.

Therefore

Anything to the power of 1 equals itself.So if follows that

At first this looks quite easy, but there is no value you can raise a positive value to that will give a negative value.So is impossible.

### Adding Logs

Proof

Let and

and

You can show that in the same way

Examples

Simplify the following

These are straight forward logs with the same base so we can join them with multiplication so the answer is

(if a base is not defined then any base will do and most people will just use base 10)We can assume they have a base, because a base is not mentioned.

So the answer is

These are straight forward logs, but they have different bases so we can not join them simply

### Multiple logs (Powers)

Consider

Using the addition rule it also equals

So

Examples

Simplify

So using the addition rule

SolveSo using the subtraction rule we get

### Solving equations with variable powers

We can solve equations with variable powers by:

- isolating the term with the variable
- taking logs of both sides
- using the multiple logs rule to take the power out of the log
- dividing by the log and solving

Examples

Solve

the is already isolated

taking logs of both sides gives

using the multiple logs rule gives

dividing gives

Solve subtracting 9 from both sides gives

SolveTaking logs and using the power rule gives