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## Algebra 2

### Unit 8: Lesson 4

The change of base formula for logarithms

# Logarithm properties review

Review the logarithm properties and how to apply them to solve problems.

## What are the logarithm properties?

Product rulelog, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis
Quotient rulelog, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis
Power rulelog, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis
Change of base rulelog, start base, b, end base, left parenthesis, M, right parenthesis, equals, start fraction, log, start base, a, end base, left parenthesis, M, right parenthesis, divided by, log, start base, a, end base, left parenthesis, b, right parenthesis, end fraction

## Rewriting expressions with the properties

We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.
For example, we can use the product rule to rewrite log, left parenthesis, 2, x, right parenthesis as log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis. Because the resulting expression is longer, we call this an expansion.
In another example, we can use the change of base rule to rewrite start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log, start base, 2, end base, left parenthesis, x, right parenthesis. Because the resulting expression is shorter, we call this a compression.
Problem 1
Expand log, start base, 2, end base, left parenthesis, 3, a, right parenthesis.

Want to try more problems like this? Check out this exercise.

## Evaluating logarithms with calculator

Calculators usually only calculate log (which is log base 10) and natural log (which is log base e).
Suppose, for example, we want to evaluate log, start base, 2, end base, left parenthesis, 7, right parenthesis. We can use the change of base rule to rewrite that logarithm as start fraction, natural log, left parenthesis, 7, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction and then evaluate in the calculator:
\begin{aligned} \log_2(7)&=\dfrac{\ln(7)}{\ln(2)} \\\\ &\approx 2.807 \end{aligned}
Problem 1
Evaluate log, start base, 3, end base, left parenthesis, 20, right parenthesis.