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MathidiaMathematics Academy
Exponentials and logarithms
Lesson 28 of 75
Standard practice

Laws of logarithms

Objective. Combine and expand logarithmic expressions with restrictions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The laws of logarithms let you combine or expand expressions involving logs, mirroring the index laws because logarithms are powers. The three key rules are logb(xy) = logb(x) + logb(y), logb(x/y) = logb(x) − logb(y), and logb(xⁿ) = n×logb(x). For example log2(8×4) = log2(8) + log2(4) = 3 + 2 = 5, which checks since 8×4 = 32 = 2⁵. These laws require all arguments to be positive, and logb(1) = 0 while logb(b) = 1 for any valid base b. Expanding a single complicated log into several simpler ones, or combining several into one, uses these three rules in either direction.

Worked example

Use the laws of logarithms to write log2(40) − log2(5) as a single logarithm and evaluate it.

2 … 40
Illustrated card showing 2 … 40
  1. 1Apply the subtraction law: log2(40) − log2(5) = log2(40/5).
  2. 2Simplify the fraction: 40/5 = 8.
  3. 3So the expression equals log2(8).
  4. 4Since 2³ = 8, log2(8) = 3.

Answer: 3

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Common mistake. Students wrongly simplify log(x) + log(y) as log(x + y) instead of log(xy). Always remember addition of logs corresponds to multiplication inside, not addition inside, the logarithm.

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