Condense Logarithms Calculator - Combine Log Expressions

Condensing logarithms means rewriting a sum, difference, or scalar multiple of logarithms with the same base as a single logarithm. The technique relies on three classical identities — the product rule log_b(a) + log_b(c) = log_b(a·c), the quotient rule log_b(a) − log_b(c) = log_b(a/c), and the power rule k·log_b(a) = log_b(a^k). Together with the change-of-base formula, these three rules let you manipulate any logarithmic expression that involves a common base. The calculator accepts symbolic inputs such as x, (x + 1), or 5 because condensing is fundamentally a symbolic operation: the result is an expression, not a numeric value. Choose the operation that matches your problem — Addition for log_b(a) + log_b(b), Subtraction for log_b(a) − log_b(b), or Power for k·log_b(a) — and the calculator will assemble the corresponding condensed form. The base selector covers the three most common cases (10, e, and 2) and offers a custom base for any positive number not equal to 1. Why bother condensing? In calculus, a single logarithm is far easier to differentiate or integrate than a long sum of logarithms. When solving logarithmic equations, condensing the left-hand side lets you cancel the log against an exponential to recover a polynomial equation. In data analysis, condensing log-likelihoods into a single log-product simplifies maximum likelihood computations. In information theory, condensing terms involving log_2 reveals entropy and mutual information in their cleanest form. A few important caveats. All logarithms in a single condensing step must share the same base — you cannot combine log_2(x) with log_10(y) without first using the change-of-base formula. The arguments of every logarithm must be positive in the real-number setting; if you allow zero or negative arguments, the equations become identities only on a restricted domain. The power rule applies the exponent k to the argument of the log, not to the log itself: k·log_b(a) becomes log_b(a^k), never (log_b(a))^k. Use the condense logarithms calculator whenever you need to simplify a homework problem in algebra or precalculus, prepare an expression for differentiation in calculus, or check the work of a step in a longer derivation.