Properties of logarithms (evaluate / transform)
Properties of Logarithms
Properties of Logarithms is a topic in Exponential & Logarithmic Functions in the Common Core State Standards. It is aligned to Standard F-LE.4, which requires students to use the properties of logarithms to evaluate and transform expressions.
The properties of logarithms — the product, quotient, and power rules, plus change of base — are used to expand and condense logarithmic expressions.
Theory
Because logarithms are exponents, they inherit the exponent rules — turned into rules for combining logs:
- Product: \(\log_b(MN)=\log_b M+\log_b N\).
- Quotient: \(\log_b\dfrac{M}{N}=\log_b M-\log_b N\).
- Power: \(\log_b(M^p)=p\log_b M\).
The change-of-base formula lets you compute any log with the \(\ln\) or \(\log\) on your calculator.
The four properties:
How to use the log properties
- To expand: quotient \(\to\) product \(\to\) power, working outward.
- To condense: power rule first (move coefficients to exponents), then product/quotient.
- To evaluate an odd base: apply change of base and use a calculator.
Apply the quotient, product, and power rules in turn.
| \(=\) | \(\log(x^2 y)-\log z\) | |
| \(=\) | \(\log x^2+\log y-\log z\) | |
| \(=\) | \(2\log x+\log y-\log z\) |
Power rule first, then quotient rule.
| \(3\log x-\log y\) | \(=\) | \(\log x^3-\log y\) |
| \(=\) | \(\log\dfrac{x^3}{y}\) |
Use \(\log_b x=\dfrac{\ln x}{\ln b}\).
| \(\log_2 50\) | \(=\) | \(\dfrac{\ln 50}{\ln 2}\) |
| \(\approx\) | \(5.64\) |
Quotient rule combines them.
| \(\log_5 100-\log_5 4\) | \(=\) | \(\log_5\dfrac{100}{4}\) |
| \(=\) | \(\log_5 25=2\) |
Common pitfalls
Frequently asked questions
What are the properties of logarithms?
Product: \(\log(MN)=\log M+\log N\); quotient: \(\log(M/N)=\log M-\log N\); power: \(\log(M^p)=p\log M\).
What is the change-of-base formula?
\(\log_b x=\dfrac{\ln x}{\ln b}\) (or with common logs), letting you compute any base on a calculator.
Can you split log(M + N)?
No. There is no rule for the log of a sum; the properties apply only to products, quotients, and powers.
What is the difference between expanding and condensing?
Expanding breaks one log into several using the rules left to right; condensing combines several into one using them right to left.