Algebra 2
Exponential and logarithmic functions
Laws of logarithms
20 practice questions
0 video lessons
Theory + worked examples
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: \(\log_b(m^p)=p\log_b m\).
Change of base: \(\log_b x=\dfrac{\log x}{\log b}\) lets a calculator evaluate any log.
The three laws of logarithms.
Change of base.
The laws:
\[\log_b(mn)=\log_b m+\log_b n,\qquad \log_b(m^p)=p\log_b m\]
These only combine logs of the same base.
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.
Example 1 — Expand
Expand \(\log(xy^2)\).
Solution
Use the product and power laws.
| \(\log(xy^2)\) | \(=\) | \(\log x+\log y^2\) |
| \(=\) | \(\log x+2\log y\) |
Example 2 — Condense
Write \(2\log x-\log y\) as one logarithm.
Solution
Power law, then quotient law.
| \(2\log x-\log y\) | \(=\) | \(\log x^2-\log y\) |
| \(=\) | \(\log\dfrac{x^2}{y}\) |
Example 3 — Evaluate with laws
Evaluate \(\log_2(8\cdot4)\).
Solution
Split with the product law.
| \(\log_2 8+\log_2 4\) | \(=\) | \(3+2\) |
| \(=\) | \(5\) |
Example 4 — Change of base
Write \(\log_2 10\) using base-10 logs.
Solution
Apply the change-of-base formula.
| \(\log_2 10\) | \(=\) | \(\dfrac{\log 10}{\log 2}\) |
| \(\approx\) | \(3.32\) |
Common pitfalls
\(\log(m+n)\neq\log m+\log n\) — only products split.
The power law brings the exponent to the front, not the argument.
Same base required to combine logs.
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.
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Logarithms (definition and evaluation)
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Natural logarithm and the number e
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