Condense Logarithms Calculator

Background.

Logarithms were invented to turn multiplication into addition. After John Napier published his tables in 1614, astronomers could multiply two long numbers by looking up their logarithms, adding them, and reading the product back from the table, and every slide rule built afterwards did the same with sliding scales. The rule behind that trick, log A + log B = log(AB), is the one at work when you condense an expression, together with its partner for division, log A − log B = log(A/B).

Condensing runs those rules in the opposite direction to expanding. Instead of splitting one logarithm into several, you gather a sum and difference of logarithms into a single one, and that is usually the step that unlocks a logarithmic equation: once a side is one log, it can be rewritten in exponential form and solved.

The page handles the most common shape, log A + log B − log C, and returns the single argument AB/C. All three logarithms must share a base, but which base does not matter — the condensed argument is the same for base 10, base e or base 2. The arguments must be positive, because a real logarithm of zero or a negative number does not exist. Coefficients other than 1 need the power rule first, and expressions containing variables have to be condensed by hand; the questions below show how to fit both kinds into the three fields.

What is condense logarithms calculator?

Condensing logarithms means rewriting a sum or difference of logarithms with the same base as one logarithm. The product rule turns added logarithms into the logarithm of a product, and the quotient rule turns a subtracted logarithm into division, so log_b A + log_b B − log_b C = log_b(AB/C). The output here is that combined argument, AB/C, a pure number; its logarithm in your chosen base equals the original expression.

How to use this calculator.

1. 01Enter the arguments of the two added logarithms in the first and second fields. If only one logarithm is added, put 1 in the second field — log 1 is zero, so it changes nothing.
2. 02Enter the argument of the subtracted logarithm in the third field, or 1 if nothing is subtracted.
3. 03Apply the power rule before entering any term with a coefficient: 2 log 3 becomes log 9, and ½ log 16 becomes log 4.
4. 04Read the result as the argument of one logarithm in the same base as the original terms, so log 8 + log 9 − log 6 condenses to log 12.

The formula.

Two identities do the work. The product rule, log_b A + log_b B = log_b(AB), holds because logarithms are exponents: if b^x = A and b^y = B, then b^(x+y) = AB. The quotient rule follows the same way — with b^z = C, b^(x+y−z) = AB/C — so log_b A + log_b B − log_b C = log_b(AB/C). The page computes AB/C directly, multiplying the first two arguments and dividing by the third. The base b never enters that arithmetic, which is why a single answer serves every base. The result grows in proportion to A and to B and shrinks in proportion to C, so doubling C halves it. Each argument must be positive, and the fields accept values from 0.000001 to 1,000,000,000.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.