Laws of logarithms
Laws of Logarithms
Laws of Logarithms is a topic in Exponential & Logarithmic Functions in the Common Core State Standards. It is aligned to Standard F-BF.5, which requires students to use the inverse relationship and properties of logarithms to rewrite and evaluate logarithmic expressions.
The laws of logarithms turn products into sums, quotients into differences, and powers into products, with change of base to evaluate any log.
Theory
The laws of logarithms mirror the exponent rules:
- 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 laws:
How to use the laws
- To expand, split products/quotients and drop exponents in front.
- To condense, reverse: coefficients become exponents.
- Keep all logs in the same base.
- Use change of base to evaluate numerically.
Use the product and power laws.
| \(\log(xy^2)\) | \(=\) | \(\log x+\log y^2\) |
| \(=\) | \(\log x+2\log y\) |
Power law, then quotient law.
| \(2\log x-\log y\) | \(=\) | \(\log x^2-\log y\) |
| \(=\) | \(\log\dfrac{x^2}{y}\) |
Split with the product law.
| \(\log_2 8+\log_2 4\) | \(=\) | \(3+2\) |
| \(=\) | \(5\) |
Apply the change-of-base formula.
| \(\log_2 10\) | \(=\) | \(\dfrac{\log 10}{\log 2}\) |
| \(\approx\) | \(3.32\) |
Common pitfalls
Frequently asked questions
What is the product law of logarithms?
\(\log_b(mn)=\log_b m+\log_b n\).
Does \(\log(m+n)=\log m+\log n\)?
No — that is a common mistake; only products split.
What is the power law?
\(\log_b(m^p)=p\log_b m\).
What is change of base?
\(\log_b x=\dfrac{\log x}{\log b}\), letting you use any base.