Product Rule

The logarithmic product rule states that

\[ log_x(a \times b) = log_x(a) + log_x(b) \]

You can decompose the multiplication of two inputs to a log function by adding the two individual logs together. The log functions must share the same base for this to work.

The logarithm exhibits this behavior because a \( log \) is simply a way of rewriting the exponent and so any product of two numbers expressed as exponents is additive.

\[ 2^2 * 2^4 = 2^6 \]

The exponents can isolated using \( log \) notation, and the above expression rewritten.

\[ log_2(4) + log_2(16) = log_2(64) = log_2(4 * 16) \]

You can write any log as a composition of other logs.

\[ ln(20) = ln(10) + ln(2) \]

\[ log_2(17) = log_2(8.5) + log_2(8.5) \]

\[ log_3(10) = log_3(3) + log_3(3 \frac{1}{3}) \]

Time Interpretation

We can use the time interpretation of logarithms to clarify why this is the case.

You can interpret the multiplication of log inputs as growing for a set amount of time and then growing again. For example, if we want to calculate the time to grow by a factor of \( 16 \) we can just use \( ln(16) \) but we can also break this down into \( ln(4) + ln(4) \). What this means intuitively is that we grow four times for some period and then we grow another four times resulting in a total \( \times 16 \) growth over two equal time periods.