About

Condensing logarithms means rewriting a sum or difference of multiple logarithmic terms as one single logarithm. The process reverses the expansion rules taught in algebra and precalculus. Three properties govern the transformation: the Power Rule moves a coefficient into the exponent (k ⋅ log_b(x) = log_b(x^k)), the Product Rule merges addition into multiplication, and the Quotient Rule converts subtraction into division. Errors in this process typically cascade through exam solutions and engineering derivations. A misplaced sign flips division to multiplication inside the argument, producing an answer orders of magnitude off. This calculator applies each rule in the canonical order and displays every intermediate step so you can trace exactly where each coefficient and operator was resolved.

The tool assumes all terms share the same base. If bases differ, the expression cannot be condensed into a single logarithm without a change-of-base conversion. Arguments must be positive real numbers, and the base must be positive and not equal to 1. Fractional coefficients and negative coefficients (subtracted terms) are fully supported. The result is given in simplified symbolic form with fractions reduced via GCD.

Formulas

The condensing process follows a strict order of operations. First, apply the Power Rule to every term. Then combine all terms using the Product and Quotient Rules.

Step 1 - Power Rule:

k ⋅ log_b(x) → log_b(x^k)

Step 2 - Product & Quotient Rules:

log_b(A) + log_b(B) − log_b(C) = log_b(A ⋅ BC)

Where k = coefficient of the logarithmic term, b = common base (must be identical across all terms), x = argument of the logarithm (must be positive), A, B = arguments of added (positive-coefficient) terms after the power rule, C = arguments of subtracted (negative-coefficient) terms after the power rule. Fractional exponents are preserved symbolically. For example, 12 ⋅ log(x) becomes log(x^1/2) which equals log(√x).

Reference Data

Rule	Expanded Form	Condensed Form	Condition
Power Rule	k ⋅ log_b(M)	log_b(M^k)	M > 0
Product Rule	log_b(M) + log_b(N)	log_b(M ⋅ N)	M, N > 0
Quotient Rule	log_b(M) − log_b(N)	log_b(MN)	M, N > 0
Change of Base	log_b(M)	log_a(M)log_a(b)	a, b > 0, ≠ 1
Log of 1	log_b(1)	0	Any valid b
Log of Base	log_b(b)	1	b > 0, ≠ 1
Log of Power of Base	log_b(b^k)	k	b > 0, ≠ 1
Natural Log (ln)	ln(M)	log_e(M)	e ≈ 2.71828
Common Log	log(M)	log₁₀(M)	Convention
Inverse: Exponentiation	b^log_b(M)	M	M > 0
Reciprocal Argument	log_b(1M)	−log_b(M)	M > 0
Reciprocal Base	log_1b(M)	−log_b(M)	b > 0, ≠ 1

Frequently Asked Questions

Condensing into a single logarithm requires all terms to share the same base. If bases differ, you must first apply the change-of-base formula: log_b(M) = log_a(M) / log_a(b) to convert all terms to a common base before condensing. This calculator flags mismatched bases as an error since mixing bases without explicit conversion produces incorrect results.

Yes. A fractional coefficient like 1/2 · log(x) becomes log(x^1/2), equivalent to log(√x). A negative coefficient like −3 · log(x) becomes log(x⁻³) = log(1/x³). The calculator handles both cases symbolically and reduces fractions via GCD.

The Power Rule must be applied before the Product/Quotient Rules. However, within the combination step, addition is commutative and the order of multiplied numerator terms does not change the result. The sign (+ or −) determines whether an argument goes to the numerator or denominator of the condensed fraction.

Logarithms are undefined for non-positive arguments in the real number system. If any term has an argument ≤ 0, the calculator rejects it with a validation error. Similarly, the base must satisfy b > 0 and b ≠ 1. These constraints are enforced before computation begins.

A term with coefficient 0 evaluates to 0 · log_b(x) = 0, which is simply the constant zero. The calculator drops such terms from the condensing process since they contribute nothing to the final expression. If all coefficients are zero, the result is 0.

Yes. Natural logarithm ln is log base e (≈ 2.71828). Select "e" as the base and the output will be displayed using ln notation. All condensing rules apply identically regardless of whether the base is 10, e, 2, or any other valid value.