Properties of logarithms (evaluate / transform)
Properties of Logarithms
Properties of Logarithms is a topic in Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.5(G), 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.