Laws of logarithms for expanding and condensing logs

A logarithm tells us which exponent gives a particular number. The laws of logarithms let us use that relationship to take an expression apart or put several terms together. We’ll go through the three main rules, then use them in two examples.

Start with the base and argument

If $a^{k} = y$ , then $\log_{a} (y) = k$ . For real logarithms, the base $a$ is positive and different from $1$ , and the argument $y$ is positive. In the rules below, both $x$ and $y$ are positive and every log uses the same base.

Use the product, quotient, and power rules

The product rule turns multiplication inside a log into addition outside it. Read the equation from right to left to combine a sum of two logs with the same base.

\log_{a} (x y) = \log_{a} (x) + \log_{a} (y)

The quotient rule turns division into subtraction. The log of the numerator comes first, so the order carries over when we condense the difference.

\log_{a} (\frac{x}{y}) = \log_{a} (x) - \log_{a} (y)

The power rule moves the exponent on the argument in front of the log. In reverse, a coefficient multiplying a log becomes a power on its argument.

\log_{a} (x^{k}) = k \log_{a} (x)

Combine a sum and evaluate the result

Both logs have base $4$ , so multiply their arguments and keep base $4$ . Since $4^{5} = 1024$ , the combined logarithm equals $5$ .

\begin{array}{l} \log_{4} (64) + \log_{4} (16) \\ = \log_{4} (64 \cdot 16) \\ = \log_{4} (1024) = 5 \end{array}

Condense a difference to a single logarithm

Move the coefficient $2$ onto the argument $5$ as an exponent. Now the two logs have the same base and we can divide their arguments.

\begin{array}{l} \log_{3} (14) - 2 \log_{3} (5) \\ = \log_{3} (14) - \log_{3} (25) \\ = \log_{3} (\frac{14}{25}) \end{array}

The exact answer is $\log_{3} (14 / 25)$ . Its value is approximately $- 0.5278$ . That sign makes sense because the argument is between $0$ and $1$ and the base is greater than $1$ .

Laws of logarithms for expanding and condensing logs

Start with the base and argument

Use the product, quotient, and power rules

Combine a sum and evaluate the result

Condense a difference to a single logarithm

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