Condense Logarithms Calculator (combine into a single logarithm)

Understanding the condense logarithms calculator — concept and principle

This calculator combines a sum and difference of logarithms into a single logarithm, using the power, product and quotient rules. It shows the numerator and denominator inside the result and lists the rules used. Use it for algebra and precalculus exercises or to simplify an expression before solving an equation.

Coefficients become exponents first; then plus means multiply and minus means divide.

How it is calculated

Power rule: n · log_b(M) = log_b(Mⁿ)

Product rule: log_b(M) + log_b(N) = log_b(MN)

Quotient rule: log_b(M) − log_b(N) = log_b(M ÷ N)

First move every coefficient into an exponent, then multiply the added terms and divide by the subtracted ones.
All terms must use the same base (common log, natural log, base 2 or another base you enter), and the arguments are assumed positive.
A plain number c becomes log_b(b^c); fractional coefficients become fractional exponents, such as ½ log x = log √x.

The principle

Logarithms turn multiplication into addition: log(MN) = log M + log N, because exponents add when powers of the same base are multiplied. Condensing is that identity used backwards. The power rule follows from repeated multiplication: log(M³) = log M + log M + log M = 3 log M. Condensing is useful because an equation with a single logarithm can be solved by rewriting it in exponential form.

Worked example

Default inputs: 2 log(x) + log(y) − 3 log(z), base 10.

Power rule: 2 log x = log(x²) and 3 log z = log(z³).
Product rule: log(x²) + log(y) = log(x²·y).
Quotient rule: log(x²·y) − log(z³) = log(x²·y ÷ z³).
Condensed form: log(x²·y/z³); numerator x²·y, denominator z³; 3 terms read; rules used power rule, product rule, quotient rule.
The arguments are letters, so no numerical value is shown (—).

How to read the result

The condensed form is equal to the original only where every original argument is positive. If you put numbers in for every argument, the page also evaluates the result, which is a quick check that the two forms agree.

Common mistakes

Writing log M + log N = log(M + N).
Forgetting to apply the power rule before combining.
Putting subtracted terms in the numerator.
Combining logarithms with different bases.

Keep in mind: The rules hold only for positive arguments. Condensing can change the domain of an expression, so check solutions in the original equation.

Frequently asked questions

How do you condense logarithms?

Move coefficients into exponents, then multiply arguments of added logs and divide by those of subtracted logs.

Can I condense logs with different bases?

Not directly; convert them to the same base first.

What is log(2) + log(5)?

log(10) = 1 in base 10.

Sources: OpenStax, College Algebra 2e (2021), §6.5 Logarithmic Properties

Checked 9 Nov 2026. OpenStax, College Algebra 2e (2021), §6.5 Logarithmic Properties — product, quotient and power rules, “Condensing Logarithmic Expressions”.

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