Logarithm Properties Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

easy
Expand log⁡(a2b5)\log\left(\frac{a^2}{b^5}\right).

Solution

  1. 1
    Apply the quotient rule: log⁡(a2)−log⁡(b5)\log(a^2) - \log(b^5).
  2. 2
    Apply the power rule: 2log⁡a−5log⁡b2\log a - 5\log b.

Answer

2log⁡a−5log⁡b2\log a - 5\log b
The quotient rule converts division inside a log into subtraction, and the power rule moves exponents to coefficients.

About Logarithm Properties

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

Learn more about Logarithm Properties →

More Logarithm Properties Examples

Example 1 easy

Expand [formula] using logarithm properties.

Example 2 medium

Condense [formula] into a single logarithm.

Example 3 medium

Use the change of base formula to evaluate [formula] to three decimal places.

← All Logarithm Properties Examples Logarithm Properties Concept