Condense Logarithms Calculator
Condense two logarithms into one using the product, quotient, and power rules, so log A + log B becomes log(AB) and log A - log B becomes log(A/B).
Condense Logarithms Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: log(A) + log(B) = log(A×B) | log(A) - log(B) = log(A/B) | n×log(A) = log(Aⁿ)
Worked example — 5
Formula
log(A) + log(B) = log(A×B) | log(A) - log(B) = log(A/B) | n×log(A) = log(Aⁿ)
Logarithm properties allow combining (condensing) multiple log expressions into a single logarithm.
Worked Examples
Example 1: Addition
Problem:log(100) + log(1000) = 2 + 3
Solution:log(100×1000) = log(100000) = 5
Result:5
Frequently Asked Questions
Why condense logarithms?
Condensing simplifies expressions for solving equations. log(2)+log(3)=log(6) is simpler to work with algebraically.
What are the three main logarithm properties used for condensing?
The three fundamental logarithm properties are: (1) Product rule: log(A) + log(B) = log(A × B) — adding logs multiplies the arguments; (2) Quotient rule: log(A) - log(B) = log(A / B) — subtracting logs divides the arguments; (3) Power rule: n × log(A) = log(A^n) — a coefficient becomes an exponent inside the log. These rules work for any base logarithm, not just base 10. They are derived from the fundamental properties of exponents and form the backbone of solving logarithmic equations.
What is the difference between log (base 10) and ln (natural log)?
log typically refers to base-10 logarithm (common logarithm), meaning log(100) = 2 because 10^2 = 100. ln refers to the natural logarithm with base e (approximately 2.71828), meaning ln(e^2) = 2. In scientific and mathematical contexts, ln is more common because e arises naturally in calculus, exponential growth, and differential equations. In engineering and everyday calculations, base-10 log is often used because it aligns with the decimal number system. The change of base formula allows converting between any bases: log_b(x) = ln(x) / ln(b).
How are logarithms used in real-world applications?
Logarithms appear across many fields: in acoustics, the decibel scale uses log base 10 to measure sound intensity (10 dB increase = 10× more intense); in seismology, the Richter magnitude scale uses logarithms (a magnitude 7 earthquake is 10× more powerful than a magnitude 6); in chemistry, pH = -log[H+] measures acidity; in finance, continuously compounded interest uses the natural log; in information theory, Shannon entropy uses log base 2 to measure information content in bits; and in computing, binary search and sorting algorithms are analyzed using log base 2 to describe their time complexity.
What domain restrictions apply when I condense a logarithmic expression?
A real logarithm is defined only for a strictly positive argument, so every individual log in the original expression carries its own restriction. Condensing can hide those restrictions: log(x) + log(x - 3) becomes log(x^2 - 3x), and the condensed form is defined for x < 0 as well, even though the original requires x > 3. When you solve an equation this way, always substitute your answers back into the uncondensed expression and discard any extraneous roots.
How do I condense an expression that has coefficients in front of the logs?
Apply the power rule first to absorb each coefficient into its argument as an exponent, then use the product and quotient rules to merge what remains. For example 2 log(x) + 3 log(y) - log(z) becomes log(x^2) + log(y^3) - log(z), then log(x^2 y^3), then log(x^2 y^3 / z). Doing it in the other order is the usual source of mistakes, because a coefficient sitting outside a log cannot be combined with a neighbouring term.
Can I condense two logarithms that have different bases?
Not directly. The product and quotient rules only apply when both logarithms share the same base, because they come from the exponent laws for that base. Convert first with the change of base formula, log_b(x) = ln(x) / ln(b), so that log_2(8) + log_10(5) is rewritten with a single base before merging. If the bases cannot be reconciled neatly, evaluate each logarithm numerically and add the values instead.
What is the difference between condensing and expanding logarithms?
They are inverse manipulations of the same three rules. Expanding breaks a single logarithm of a product, quotient or power into a sum or difference of simpler logs, which is what you want when differentiating or when isolating one variable. Condensing collapses several logs back into one, which is what you want before exponentiating both sides of an equation, since a lone logarithm can be undone in a single step.
Why does log(A) + log(B) not equal log(A + B)?
Because addition of logarithms mirrors multiplication of the arguments, not addition. Taking A = 10 and B = 100, log(10) + log(100) = 1 + 2 = 3, and log(10 x 100) = log(1000) = 3, which matches the product rule. But log(10 + 100) = log(110) is about 2.041, nowhere near 3. There is no logarithm rule that simplifies the log of a sum, which is why sums inside a logarithm usually have to be factored first.
What do the two input boxes on Condense Logarithms Calculator actually represent?
You enter the values of the logarithms themselves, not the arguments. Typing 2 and 3 means log(A) = 2 and log(B) = 3, so the tool reconstructs A = 10^2 = 100 and B = 10^3 = 1000 in base 10, then applies the product or quotient rule and shows the condensed logarithm. Adding gives log(100000) = 5 and subtracting gives log(0.1) = -1. If you already have A and B, take their base-10 logs before entering them.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Expanding Logarithms Calculator
Calculate expanding logarithms with inputs, formulas, and instant results.
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.