Expanding Logarithms Calculator (product, quotient & power rules)

Understanding the expanding logarithms calculator — concept and principle

This calculator expands the logarithm of a product, quotient or power into a sum and difference of simpler logarithms, using the product, quotient and power rules. It shows the expression being expanded, the rules used and the number of log terms. Use it for algebra and precalculus exercises and before differentiating logarithms.

How it is calculated

Product rule: log_b(MN) = log_b(M) + log_b(N)

Quotient rule: log_b(M/N) = log_b(M) − log_b(N)

Power rule: log_b(Mⁿ) = n·log_b(M)

• Roots are powers: √M = M^(1/2), ∛M = M^(1/3).
• A bracket containing a sum, such as (x + 1), cannot be split, because there is no rule for log(M + N).
• Variables are assumed positive so every logarithm is defined.

The principle

A logarithm is an exponent, and exponents add when powers with the same base are multiplied: bᵐ × bⁿ = bᵐ⁺ⁿ. Taking logarithms turns that into log(MN) = log M + log N. Division subtracts exponents, giving the quotient rule, and repeated multiplication gives the power rule. Expanding is useful because sums are easier to differentiate, estimate and solve than products inside a logarithm.

Worked example

Default inputs: log(x³y²/z), base 10.

1. Logarithm being expanded: log(x^3 y^2 / z).
2. Quotient rule: log(x³y²) − log(z).
3. Product rule: log(x³) + log(y²) − log(z).
4. Power rule: 3 log(x) + 2 log(y) − log(z).
5. Rules used: product rule, quotient rule, power rule; 3 log terms; no numeric value because the factors are letters (—).

How to read the result

Every factor in the numerator becomes a plus term and every factor in the denominator a minus term, each multiplied by its exponent. Check by putting numbers in: with x = 10, y = 10, z = 10, the original is log(10⁴) = 4 and the expansion is 3 + 2 − 1 = 4.

Common mistakes

• Splitting log(x + y) into log x + log y.
• Forgetting the power rule on a root: log √x = ½ log x.
• Putting the exponent on the wrong factor, such as treating log(xy)² as 2 log x + log y.
• Ignoring the domain: the rules need positive arguments.

Frequently asked questions

How do you expand a logarithm?

Split products into sums, quotients into differences, and bring exponents to the front.

Can log(x + y) be expanded?

No, there is no rule for the logarithm of a sum.

What is ln √(x/y) expanded?

½ ln x − ½ ln y.

Sources: OpenStax, College Algebra 2e (2021), §6.5 Logarithmic Properties