Condense Logarithms Calculator — Log Rules Step by Step
Apply the power, product and quotient rules to condense c₁ log(a) ± c₂ log(b) into a single logarithm, shown step by step.
Logarithm base
Operation
log_10(128)
- 1
Power rule on first term
4 ^ 2 = 16 - 2
Power rule on second term
2 ^ 3 = 8 - 3
Product rule (×)
16 × 8 = 128log(x) + log(y) = log(x·y) - 4
Evaluate logarithm
log(128) = 2.107210
How does this calculator work?
Condensing logarithms uses three rules: the power rule moves a coefficient inside as an exponent (c log x → log xᶜ), the product rule merges a sum (log x + log y → log xy), and the quotient rule merges a difference (log x − log y → log(x/y)). Apply power rule first, then product/quotient rule, giving a single log expression.
Formula
How this is calculated
Condensing a logarithmic expression means rewriting a sum or difference of log terms as a single logarithm. Three rules make this possible, and they must be applied in the correct order.
First, the **power rule**: any coefficient in front of a logarithm moves inside as an exponent — c × log_b(x) = log_b(xᶜ). Second, the **product rule**: the sum of two logs with the same base equals the log of the product — log_b(x) + log_b(y) = log_b(x·y). Third, the **quotient rule**: the difference of two logs equals the log of the quotient — log_b(x) − log_b(y) = log_b(x/y). This calculator applies all three in sequence: it first uses the power rule to absorb each coefficient into its argument, then uses the product or quotient rule to merge the two terms into one.
All three rules require the logarithms to share the same base, which is why this calculator fixes a single base for both terms. The numerical result is computed using the change-of-base formula log_b(x) = ln(x) / ln(b) in floating-point arithmetic; the condensed algebraic form is shown alongside.
Frequently asked questions
Condensing means rewriting multiple log terms as a single log. The opposite is "expanding" (using log rules in reverse to split one log into several). Condensing is useful when solving logarithmic equations — you need a single log before taking the anti-log of both sides.
Yes. The power rule works for any real coefficient: c × log_b(x) = log_b(x^c). Fractional exponents produce roots — for example, (1/2) × log(x) = log(√x).
The product and quotient rules only work when the bases are identical. To combine logs of different bases, first use the change-of-base formula to convert both to the same base, then condense.
Also known as
TG we-Calculate Editorial Team. (2026). Condense Logarithms Calculator — Log Rules Step by Step [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/condense-logarithms-calculator
TG we-Calculate Editorial Team. "Condense Logarithms Calculator — Log Rules Step by Step." TG we-Calculate. 2026. https://we-calculate.com/calculator/condense-logarithms-calculator.
TG we-Calculate Editorial Team, "Condense Logarithms Calculator — Log Rules Step by Step," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/condense-logarithms-calculator
@misc{wecalculate_condense_logarithms_calculator, title = {Condense Logarithms Calculator — Log Rules Step by Step}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/condense-logarithms-calculator}}, year = {2026}, note = {TG we-Calculate} }
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