Combining Product Rule and Quotient Rule in Logarithms

Once you know the individual log rules, most real problems don't hand you a clean single product or a clean single quotient. Instead, you get an expression with multiplication and division tangled together inside one logarithm, such as \( \log\left(\dfrac{xy}{z}\right) \). To break this apart (or to reverse the process and squeeze several logs into one), you need to apply the product rule of logarithms and the quotient rule side by side. This lesson is about handling both at once, in either direction.

Before combining anything, it helps to have both rules written out clearly. For any positive base \( b \) and positive numbers \( M \) and \( N \):

If you need a refresher on what a logarithm actually represents before working with these rules, take a look at what is a logarithm? first.

The most common task is expanding a single log that contains both a product and a quotient. The strategy is: identify the top (numerator) and bottom (denominator) separately, apply the quotient rule to split the fraction into a subtraction, and then apply the product rule to any remaining multiplication.

Example 1: Expand \( \log\left(\dfrac{xy}{z}\right) \).

Treat \( xy \) as the numerator and \( z \) as the denominator. By the quotient rule:

\( \log\left(\dfrac{xy}{z}\right) = \log(xy) - \log z \)

Now apply the product rule to \( \log(xy) \):

\( \log(xy) - \log z = \log x + \log y - \log z \)

Notice the subtraction from the quotient rule only attaches to \( \log z \), not to \( \log y \) as well — that grouping matters and is where most errors happen.

Example 2: Expand \( \log_2\left(\dfrac{a^2 b}{c \, d}\right) \).

Split into numerator over denominator first:

\( \log_2(a^2 b) - \log_2(cd) \)

Then expand each piece with the product rule:

\( \left(2\log_2 a + \log_2 b\right) - \left(\log_2 c + \log_2 d\right) \)

Distribute the negative sign across every term that came from the denominator:

\( 2\log_2 a + \log_2 b - \log_2 c - \log_2 d \)

This last step, distributing the subtraction to every factor in the denominator, is the part that combining both rules adds on top of using just one rule alone.

The reverse skill is just as important: given a sum and difference of logs, write it as one logarithm. Work from the rules in the opposite direction — added logs become multiplied arguments, subtracted logs become divided arguments.

Example 3: Condense \( \log x + \log y - \log z \).

Group the addition first: \( \log x + \log y = \log(xy) \). Then apply the quotient rule to the subtraction:

\( \log(xy) - \log z = \log\left(\dfrac{xy}{z}\right) \)

Example 4: Condense \( 2\log_3 a + \log_3 b - \log_3 c - \log_3 d \).

First rewrite \( 2\log_3 a \) as \( \log_3(a^2) \) using the power rule, then combine the added terms into a product and the subtracted terms into a division:

\( \log_3(a^2) + \log_3 b - \log_3 c - \log_3 d = \log_3\left(\dfrac{a^2 b}{cd}\right) \)

Every term that was added ends up on top of the fraction, and every term that was subtracted ends up on the bottom.

When both rules appear in the same problem, three errors show up more than any others:

• Forgetting that subtraction from the quotient rule must be distributed to every term that came out of the denominator, not just the first one.
• Mixing up which rule to apply first: always deal with the outer structure (is it a product or a quotient overall?) before expanding what's inside each piece.
• Trying to combine \( \log x + \log y \) into \( \log(x+y) \) instead of \( \log(xy) \). Logs never distribute over plain addition inside their argument.

These skills feed directly into solving logarithmic equations, where condensing multiple logs into one is often the first step before removing the log altogether.

When you see a log expression that combines multiplication and division, slow down and label the numerator and denominator before touching anything. Apply the quotient rule once to split the fraction, then apply the product rule to any remaining products on either side. For condensing, do the reverse: handle exponents with the power rule first, then merge sums into products and differences into quotients, one step at a time rather than all at once.

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