A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

The Laws of Logarithms

A1.4About 70 minutes

Learning goals

I can state the product, quotient and power laws of logarithms.
I can explain how each law of logarithms comes from a law of exponents.
I can verify the laws numerically with a calculator.
I can recognize common false "laws", such as log (M + N) = log M + log N, and disprove them with a counterexample.

Words to know

product law
log_b (MN) = log_b M + log_b N, for M, N > 0. The log of a product is the sum of the logs.

quotient law
log_b (M/N) = log_b M − log_b N, for M, N > 0. The log of a quotient is the difference of the logs.

power law
log_b (Mᵖ) = p log_b M, for M > 0 and any real p. The exponent comes down in front as a multiplier.

counterexample
One example that shows a general statement is false.

Logs turn hard operations into easy ones

Back in lesson 1 you saw a double number line: when x is multiplied by 2, log₂ x goes up by 1. And in lesson 2 the slide rule multiplied numbers by adding lengths. Both pictures point to the same idea: logarithms turn multiplication into addition. For centuries this was the main reason people used them, because adding long numbers by hand is much easier than multiplying them.

In this lesson you will prove three laws of logarithms. Each one is an exponent law in disguise, because a logarithm is an exponent.

Remember from Grade 11: the exponent laws for the same base b:

bᵐ × bⁿ = b^(m + n) (multiply powers: add exponents)
bᵐ ÷ bⁿ = b^(m − n) (divide powers: subtract exponents)
(bᵐ)ᵖ = b^(mp) (power of a power: multiply exponents)

The product law

Let M and N be positive, and let m = log_b M and n = log_b N.

In exponential form, M = bᵐ and N = bⁿ.

Multiply: MN = bᵐ × bⁿ = b^(m + n).

In logarithmic form, that says log_b (MN) = m + n. Substitute back:

log_b (MN) = log_b M + log_b N

The quotient law

With the same M = bᵐ and N = bⁿ: M/N = bᵐ ÷ bⁿ = b^(m − n). So:

log_b (M/N) = log_b M − log_b N

The power law

With M = bᵐ, raise both sides to the power p: Mᵖ = (bᵐ)ᵖ = b^(mp). So log_b (Mᵖ) = mp = p × m:

log_b (Mᵖ) = p log_b M

The power law works for any real p, including fractions and negatives. Two useful special cases: log_b (1/N) = log_b (N⁻¹) = −log_b N, and log_b ⁿ√M = log_b (M^(1/n)) = (1/n) log_b M.

Each logarithm law is an exponent law read in logarithmic form: logs are exponents, so they add, subtract and multiply just like exponents do.

Worked example

Example 1: Verify the laws with a calculator

Use common logarithms (to four decimal places) to check: (a) log 6 = log 2 + log 3, (b) log 5 = log 10 − log 2, (c) log 8 = 3 log 2.

(a) log 2 + log 3 ≈ 0.3010 + 0.4771 = 0.7781, and log 6 ≈ 0.7782. They agree (the last digit differs only because of rounding).
(b) log 10 − log 2 ≈ 1 − 0.3010 = 0.6990, and log 5 ≈ 0.6990 ✓
(c) 3 log 2 ≈ 3(0.30103) = 0.90309, and log 8 ≈ 0.90309 ✓

All three laws check out numerically. This is the "verify" part of the curriculum: a calculator cannot prove a law (that needs the algebra above), but it is a great way to catch a mistake.

log₂ 8 + log₂ 4 = 3 + 2 = 5, and log₂ (8 × 4) = log₂ 32 = 5. Adding the logs gives the log of the product.

Worked example

Example 2: Evaluate with the product and quotient laws

Evaluate exactly: (a) log₆ 4 + log₆ 9, (b) log₃ 54 − log₃ 2, (c) log 50 + log 2.

(a) Product law: log₆ (4 × 9) = log₆ 36 = 2.
(b) Quotient law: log₃ (54 ÷ 2) = log₃ 27 = 3.
(c) Product law: log (50 × 2) = log 100 = 2.
Notice that none of the separate logs is a nice number (log₆ 4 ≈ 0.774), but the combinations are.

(a) 2 (b) 3 (c) 2

Worked example

Example 3: Evaluate with the power law

Evaluate exactly: (a) log₂ 8⁵, (b) log₅ √125, (c) log₄ (1/64).

(a) log₂ 8⁵ = 5 log₂ 8 = 5 × 3 = 15.
(b) √125 = 125^(1/2), so log₅ √125 = (1/2) log₅ 125 = (1/2)(3) = 3/2.
(c) log₄ (1/64) = −log₄ 64 = −3.

(a) 15 (b) 3/2 (c) −3

Laws that do NOT exist

The laws above are about the logarithm of a product, quotient or power. There is no law for the logarithm of a sum or difference, and a quotient of logs is not the log of a quotient. Watch out for these tempting mistakes:

log (M + N) ≠ log M + log N
log (M − N) ≠ log M − log N
(log M) ÷ (log N) ≠ log (M/N). (In fact, log M ÷ log N = log_N M, by change of base.)
(log M)ᵖ ≠ p log M. The power law needs the exponent inside: log (Mᵖ).

Worked example

Example 4: Disprove with a counterexample

Show that log (M + N) = log M + log N is false in general.

Try M = N = 10.
Left side: log (10 + 10) = log 20 ≈ 1.301.
Right side: log 10 + log 10 = 1 + 1 = 2.
1.301 ≠ 2, so the statement is false.

One counterexample is enough to show it is not a law. (The right side actually equals log (MN) = log 100 = 2.)

Which law?

Look at what is inside the logarithm. Product inside → sum of logs. Quotient inside → difference of logs. Power inside → multiply by the exponent. Sum inside → no law; leave it alone.

Just for fun

Logarithms turn multiplication into addition, division into subtraction and powers into multiplication. They are basically the friend who offers to carry the heaviest box for you.

Hands-on activity

Law or lie?

You need: a scientific calculator, 10 index cards and a pencil. A partner is helpful.

Write each of these statements on its own card: log (2 × 7) = log 2 + log 7; log (2 + 7) = log 2 + log 7; log (20 ÷ 4) = log 20 − log 4; log 20 ÷ log 4 = log 5; log 3⁴ = 4 log 3; (log 3)⁴ = 4 log 3; log √10 = 1/2; log 1 = 0; log (1/5) = −log 5; log 100 − log 10 = log 90.
Shuffle. Take turns drawing a card. Before calculating, predict: law (always true) or lie (false)?
Check with the calculator, and for each lie write what the right side really equals.

What to look for: The lies are log (2 + 7) = log 2 + log 7 (0.954 vs 1.146), log 20 ÷ log 4 = log 5 (2.161 vs 0.699), (log 3)⁴ = 4 log 3 (0.052 vs 1.908) and log 100 − log 10 = log 90 (1 vs 1.954). All the others are true.

Your turn

Write log₂ (5 × 7) as a sum of logs.

Write log (x/3) as a difference of logs (x > 0).

Write log₅ 7³ using the power law.

Evaluate log₁₂ 3 + log₁₂ 48.

Evaluate log₂ 80 − log₂ 5.

Evaluate log₃ 9⁴.

Give a counterexample to show log (M − N) = log M − log N is false.

More practice: warm-up

Which law says log_b (MN) = log_b M + log_b N?

Evaluate log 4 + log 25.

Evaluate log₂ 24 − log₂ 3.

Evaluate log₇ 49⁵.

Evaluate log₅ (1/25).

More practice: on your own

Evaluate log₄ 2 + log₄ 32.

Evaluate log₂ ∛16.

Evaluate log 0.5 + log 20 + log 10.

Prove the quotient law: if M = bᵐ and N = bⁿ, show log_b (M/N) = log_b M − log_b N.

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, find log 18 without the LOG key.

Challenge

Use the laws to show that log_b 1 = 0 without using b⁰ = 1. (Hint: 1 = M/M.)

Use the power law to prove the change of base formula log_b x = log x ÷ log b.

Is log (M²) = (log M)² ever true? Find all positive M for which it is.

Check your understanding

Quiz

Which expression equals log_b (MN)?

Evaluate log₂ 40 − log₂ 5.

Which expression equals log 7⁵?

Which statement is FALSE for positive M and N?

Evaluate log 8 + log 125.

Which exponent law gives the power law of logarithms?

For parents and teachers

Big idea (A1.4): because a logarithm is an exponent, the three exponent laws become three logarithm laws: product → sum, quotient → difference, power → multiple.

How to help: Ask your teen to explain where the product law comes from, in their own words ("when you multiply powers you add exponents, and logs are exponents"). Then play the "Law or lie?" game from the activity. Spotting false laws is as important as knowing true ones.

Common difficulties:

Inventing a law for log (M + N). There is none.
Confusing log M ÷ log N with log (M/N).
Applying the power law to (log M)², which is a square of a log, not a log of a square.
Forgetting that the laws need positive arguments: log (−2)(−5) is log 10, but log (−2) + log (−5) is undefined.