As you have probably noticed, your calculator only has two log buttons: the common log, log _{10} and the natural log, log _{e} or ln. Fortunately, by using the change of base theorem, you can evaluate a log of any base.

### Change of Base Theorem

### Evaluate

Round the result to three decimal places.

log _{6} 9

Take the log of 9 divided by the log of 6. The result is 1.226.

log _{2} 14

Divide the log of 14 by the log of 2 The answer is 3.807.

log _{1/4} 12

You can also use the natural log instead of the common log. Take ln 12 / ln .25. The result is -1.792.

### Evaluate

Do not use a calculator.

## Properties of Logarithms

Because the answer to a logarithmic equation is the exponent in an exponential equation, it makes sense that logarithms should behave as exponents do.

### Expand

log _{5} 3x^{2}y

### Condense

3 ln(x – 5) + ln x

Precalculus with Limits – Larson & Hostetler p.239-245