Logarithm Properties
Essential logarithm rules including product, quotient, power, and change of base.
Learn how to simplify and solve log expressions.
The Formulas
Product Rule: log(ab) = log(a) + log(b)
Quotient Rule: log(a/b) = log(a) - log(b)
Power Rule: log(aⁿ) = n × log(a)
Change of Base: log_b(a) = log(a) / log(b)
Logarithms are the inverse of exponents.
If b^x = a, then log_b(a) = x. These properties help simplify complex log expressions.
Variables
| Symbol | Meaning |
|---|---|
| log | Logarithm (base 10 unless otherwise noted) |
| ln | Natural logarithm (base e ≈ 2.718) |
| log_b | Logarithm with base b |
| a, b | Positive real numbers |
| n | Any real number (exponent) |
Additional Properties
- log_b(1) = 0 — because b⁰ = 1 for any base
- log_b(b) = 1 — because b¹ = b
- log_b(b^x) = x — log and exponent cancel out
- b^(log_b(x)) = x — exponent and log cancel out
Example 1
Simplify: log(50) + log(2)
Using the Product Rule: log(ab) = log(a) + log(b)
log(50) + log(2) = log(50 × 2) = log(100)
log(100) = 2
Example 2
Calculate log₅(20) using the change of base formula
log₅(20) = log(20) / log(5)
log(20) ≈ 1.3010, log(5) ≈ 0.6990
log₅(20) ≈ 1.3010 / 0.6990
log₅(20) ≈ 1.861
When to Use It
Use logarithm properties when:
- Simplifying expressions that involve products, quotients, or powers inside a log
- Solving exponential equations (take the log of both sides)
- Converting between different log bases (change of base formula)
- Working with scientific data that spans many orders of magnitude (pH, decibels, Richter scale)