Condense Logarithms Calculator

Combine two logarithmic terms — each with its own coefficient and argument — into a single logarithm using the power, product, and quotient rules, then see the exact numeric value.

Quick Facts

Rules used
Power, product, and quotient rules of logarithms
a·log_b(x) = log_b(x^a); log_b(M)+log_b(N) = log_b(MN); log_b(M)−log_b(N) = log_b(M/N).

Your Results

Calculated
Condensed argument
-
The number inside the single logarithm
Expression value
-
log_b(condensed argument)
Term 1 value
-
a₁ × log_b(x₁)
Term 2 value
-
a₂ × log_b(x₂)

Ready

Enter a base, two coefficient/argument pairs, and an operation, then press Calculate.

How condensing logarithms works

Condensing logarithms is the reverse of expanding them: you start with two or more logarithmic terms and rewrite them as a single logarithm. This calculator takes a two-term expression of the form a₁·log_b(x₁) ± a₂·log_b(x₂) and condenses it into one logarithm, log_b(argument), while also reporting the exact numeric value.

Formula and method

Three log rules do all the work, and they are applied in a fixed order:

  • Power rule (apply first): a·log_b(x) = log_b(xa). Each coefficient becomes an exponent on its own argument.
  • Product rule (for addition): log_b(M) + log_b(N) = log_b(M·N). Added logs become a single log of a product.
  • Quotient rule (for subtraction): log_b(M) − log_b(N) = log_b(M/N). Subtracted logs become a single log of a quotient.

Putting it together, this calculator computes the condensed argument as x₁a₁ · x₂a₂ when the terms are added, or x₁a₁ / x₂a₂ when the second term is subtracted, then evaluates log_b(argument) with the change-of-base identity log_b(n) = ln(n) / ln(b). The evaluated value always equals the original expression a₁·log_b(x₁) ± a₂·log_b(x₂), which is a quick way to confirm the condensed form is correct.

Common sources of error

  • Different bases: the product, quotient, and power rules only combine logs that already share the same base. Terms with different bases need a change-of-base step first, not a direct combination.
  • Non-positive arguments: log_b(x) is undefined for x ≤ 0, so an argument of zero or a negative number has no real result.
  • Base of 1 (or 0 or negative): the base must be positive and cannot equal 1, since 1 raised to any power is always 1 and could never produce every possible argument.

Checking your result

After condensing, the fastest check is to evaluate both sides separately: compute a₁·log_b(x₁) ± a₂·log_b(x₂) directly, then compute log_b(condensed argument), and confirm they match to several decimal places (this calculator does that automatically). If a term was added but you divided instead of multiplied the arguments, or vice versa, the two values will disagree.

Applications

Condensing logarithms is a standard algebra and precalculus step used to simplify a logarithmic equation before solving it (so both sides can be set equal to a single log and the arguments compared directly), to prepare an expression for differentiation in calculus, or simply to write a cleaner final answer on a homework problem.

Frequently Asked Questions

What does it mean to condense logarithms?
Condensing means rewriting a sum or difference of logarithmic terms as a single logarithm using three log rules: the product rule log_b(M) + log_b(N) = log_b(MN), the quotient rule log_b(M) − log_b(N) = log_b(M/N), and the power rule a·log_b(M) = log_b(M^a). For example, 2·log(4) + log(8) = log(4²) + log(8) = log(16·8) = log(128).
What are the three log rules used to condense an expression?
The power rule turns a coefficient into an exponent: a·log_b(x) = log_b(x^a). The product rule combines added logs into one: log_b(M) + log_b(N) = log_b(MN). The quotient rule combines subtracted logs into one: log_b(M) − log_b(N) = log_b(M/N). Apply the power rule first, then the product or quotient rule.
Why must the arguments of a logarithm be positive?
log_b(x) is only defined for x > 0 when the base b is positive and not equal to 1, because raising a positive base to any real exponent always produces a positive number — there is no real exponent that makes b^y equal zero or a negative number.
Does the base affect the condensed argument?
No. The product, quotient, and power rules hold for any valid base, so the condensed argument (the number inside the single logarithm) depends only on the original arguments and coefficients. Changing the base changes the numeric value of the logarithm, not the condensed argument itself.

Related calculators