Expanding Logarithms Calculator — Log Product, Quotient & Power Rules
Apply the three logarithm expansion rules — product, quotient, and power — to simplify expressions like log(x³ × y² / z) into 3·log(x) + 2·log(y) − log(z). Works with any base and shows a step-by-step derivation plus the numeric value when you supply numbers.
Expression type
Evaluated with the given numeric values
- 1
Compute log_b(x)
log(8) ÷ log(10) = 0.90309 - 2
Compute log_b(y)
log(5) ÷ log(10) = 0.69897 - 3
Compute log_b(z)
log(2) ÷ log(10) = 0.30103 - 4
Apply coefficients and signs
3 × 0.90309 + 2 × 0.69897 − 1 × 0.30103 = 3.806180
How does this calculator work?
Three rules expand any log expression: log(A·B) = log(A)+log(B) (product); log(A/B) = log(A)−log(B) (quotient); log(Aⁿ) = n·log(A) (power). Combined: log(x^m·y^n/z^p) = m·log(x)+n·log(y)−p·log(z). All rules apply to any base including ln.
Formula
How this is calculated
Logarithm expansion rewrites a complex log expression as a sum or difference of simpler ones using three core properties. The product rule states that the log of a product equals the sum of the logs: log(A·B) = log(A) + log(B). This follows from the exponent rule aˣ × aʸ = aˣ⁺ʸ, which when written in terms of logs gives log(A·B) = log(A) + log(B). The quotient rule is the mirror image: log(A/B) = log(A) − log(B), because division corresponds to subtracting exponents. The power rule allows an exponent on the argument to be pulled out as a coefficient: log(Aⁿ) = n·log(A), because raising a base to a power and then taking the log is the same as scaling the log by the power.
For a combined expression like log(x^m × y^n / z^p), all three rules are applied together: first the product and quotient rules split the terms with plus and minus signs, then the power rule brings the exponents out front, giving m·log(x) + n·log(y) − p·log(z).
All three rules apply to any consistent positive base (b > 0, b ≠ 1), including the common log (base 10), the natural log (base e ≈ 2.71828), and any other positive base. The rules are used in algebra, calculus, and information theory to simplify expressions, compute derivatives, and solve equations.
Frequently asked questions
Yes — all three rules (product, quotient, power) apply to ln exactly as they do to any other logarithm base. Set the base to e (≈ 2.71828) and the calculator uses ln. The rules follow purely from the definition of logarithms and are independent of the base.
Logarithms are only defined for positive real numbers (in the real number system). The expression log(x) asks "to what power must the base be raised to get x?" — and no real power of a positive number can produce zero or a negative number. For the numeric result to appear, all of x, y, and z must be greater than zero.
Yes — the same rules read in reverse allow condensing. For example, 3·log(x) + 2·log(y) − log(z) can be written as log(x³·y²/z). The calculator expands only, but applying the identities backwards produces the condensed form by hand.
Also known as
TG we-Calculate Editorial Team. (2026). Expanding Logarithms Calculator — Log Product, Quotient & Power Rules [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/expanding-logarithms-calculator
TG we-Calculate Editorial Team. "Expanding Logarithms Calculator — Log Product, Quotient & Power Rules." TG we-Calculate. 2026. https://we-calculate.com/calculator/expanding-logarithms-calculator.
TG we-Calculate Editorial Team, "Expanding Logarithms Calculator — Log Product, Quotient & Power Rules," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/expanding-logarithms-calculator
@misc{wecalculate_expanding_logarithms_calculator, title = {Expanding Logarithms Calculator — Log Product, Quotient & Power Rules}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/expanding-logarithms-calculator}}, year = {2026}, note = {TG we-Calculate} }
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