Log product and quotient rules
Expand or combine logarithms of products and quotients.
Log product and quotient rules is one of 3 logarithms formulas in the algebra section of this library, and it is used at high school level.
Why log product and quotient rules works
Multiplying two powers of the same base adds their exponents. A logarithm is an exponent, so the log of a product has to be the sum of the two logs. The quotient rule is the same statement for cancelling factors, where exponents subtract.
What each symbol means
$M,N$ are logarithm arguments and $b$ is the base.
Log product and quotient rules: when it holds
$M,N>0$, $b>0$, and $b\ne1$.
When it stops applying
The rule needs both arguments positive on their own. The log of the product of -2 and -8 is the log of 16, which is fine, but splitting it produces the log of -2 and the log of -8, neither of which exists over the reals. Expanding can destroy a perfectly valid expression.
Log product and quotient rules: a worked example
$\log_3(9x)=2+\log_3x$ when $x>0$.
The mistake to avoid
What people do: Splitting the log of a sum, writing the log of M plus N as the log of M plus the log of N.
Why it goes wrong: Those are different numbers. In base 10, the log of 2 plus 3 is about 0.699, while the log of 2 plus the log of 3 is about 0.778, which is really the log of 6.
Do this instead: Only a product inside the log may be split. If the inside is a sum, factor it first or leave the expression alone.
Log product and quotient rules: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $M,N$ are logarithm arguments and $b$ is the base.
- Check the conditions before substituting. $M,N>0$, $b>0$, and $b\ne1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Logarithms
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where log product and quotient rules comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about log product and quotient rules
Is the log of M over N the same as the log of M divided by the log of N?
No, and this trips up almost everyone. In base 10, the log of 100 over 10 is 1, while the log of 100 divided by the log of 10 is 2, a completely different value.
What is the matching rule for an exponent inside the log?
Bring the exponent out front as a multiplier, since a power is repeated multiplication and repeated multiplication becomes repeated addition of logs.
Why did my equation lose a solution after I expanded?
Because expanding narrowed the domain. Each piece now has to be positive on its own, so any candidate that made one piece negative gets excluded and must be checked against the original equation.
How do these rules help me solve an equation?
Condense both sides into a single logarithm first, then match the arguments or switch to exponential form. That turns a messy equation into an ordinary algebraic one.