Laws of Logarithms

Solve logarithmic equations and check solutions. Math 30-1 Alberta Grade 12.

Lesson 3.7 of Exponents & Logarithms in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Three Laws of Logarithms. Learn the three fundamental laws: Product Law (log of a product equals sum of logs), Quotient Law (log of a quotient equals difference of logs), and Power Law (exponent moves out front). Understand the critical order of operations and avoid common errors.
• Simplifying Log Expressions — Combining Into One. Master the strategy for writing multiple log terms as a single logarithm. Apply the Power Law first, then use Product and Quotient Laws to combine. Work through detailed examples from simple to complex expressions.
• Expanding Logarithms & Expressing in Terms of Given Values. Learn to expand a single complex log into a sum or difference of simpler logs. Master the reverse process: apply Quotient Law first, then Product Law, then Power Law. Express logarithms in terms of given variables by factoring strategically.
• Connecting Log Laws to Transformations & Identifying Errors. Apply log laws to recognize transformations between different forms of logarithmic functions. Learn to identify and correct common errors, especially the critical mistake of forgetting to distribute negative signs when expanding logs with products in the denominator.

Inside the lesson — a free preview

Why We Need Log Laws

Breaking Down Complex Logs. When logs have products, quotients, or powers inside, you can't evaluate them directly. The three laws let you expand or collapse these expressions. Both directions are tested on exams.

The Critical Order of Operations Rule

⚠️ Order Matters. • Coefficients must be +1 or −1 before combining logs

• If you have 2log(x) or 3log(y), use Power Law first

• Rule: Power Law → then Product or Quotient Law

Important Warnings — What Log Laws Are NOT

These are among the most common errors students make:

❌ Common Error #1. (x + y) ≠ (x) + (y) — the Product Law only applies to multiplication inside the argument, never addition.

❌ Common Error #2. (x - y) ≠ (x) - (y) — the Quotient Law only applies to division inside the argument, never subtraction.

❌ Common Error #3. ((x))/((y)) ≠ ((x)/(y)) — you cannot combine two separate logs that are being divided unless they share the same base AND you are using the change of base formula (covered later).

❌ Common Error #4. ( x)^2 ≠ (x^2) — an exponent on the entire log expression is not the same as an exponent inside the argument. The Power Law moves an exponent from inside to outside, not the other way around.

Worked example: a: Combining Logs Into a Single Logarithm

Question: Simplify _2(24) - _2(3)

Solution. Step 1: Identify

Both logs already have coefficient 1, so no Power Law needed.

Step 2: Apply Quotient Law

Subtract means divide the arguments:

_2(24) - _2(3) = _2((24)/(3)) = _2(8)

Step 3: Evaluate

What power gives 8?

2^3 = 8 → answer = 3

✓ Verify: _2((24)/(3)) = _2(8) = 3 ✓

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