Year 11 Extended · Unit 6 Logarithms and non-linear inequalities · Lesson 3 of 8

What is the logarithm of a product?

Each law of logarithms is an index law read through the definition. This lesson derives the three laws from what a logarithm is and uses them to combine, split and evaluate logarithms without a calculator.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From lesson 2: evaluate log3 81.
    Answer4.
  2. From Year 9: simplify 25 × 23.
    Answer28.
  3. From Year 9: simplify (52)3.
    Answer56.
Words used in this lesson
product law
the rule that gives the logarithm of a product from the logarithms of its factors. For example: finding log of 6 × 7 from log 6 and log 7.
quotient law
the rule that gives the logarithm of a quotient from the logarithms of its parts. For example: finding log of 20 ÷ 4 from log 20 and log 4.
power law
the rule that gives the logarithm of a power from the logarithm of its base. For example: finding log of 5 squared from log 5.
argument
the number inside a logarithm, which must be positive. For example: the 9 in log base 3 of 9.
combine
to write several logarithms as one. For example: log 2 + log 7 as a single logarithm.

Can a logarithm of a product be found from two logarithms?

A table in a school library in Dakar says that log2 64 = 6 and log2 32 = 5. A student needs the logarithm to base 2 of 64 × 32 and does not want to work out the product first.

Predict first: what would you guess for log2 (64 × 32), and why? Then use the figure: change the two powers and watch the bar for the product.

3
2
3
243the product m × n
5log of the product

Base 3: m = 9 and n = 27, so log m = 2 and log n = 3. The product is 243 and its logarithm is 2 + 3 = 5.

Show me the steps
  1. Base 2, powers 3 and 4: m = 8 and n = 16, so log m = 3 and log n = 4. The product is 128, which is 2 to the power 7, and 3 + 4 = 7.
  2. Base 4, powers 2 and 2: m = 16 and n = 16, so log m = 2 and log n = 2. The product is 256, which is 4 to the power 4, and 2 + 2 = 4.
  3. Base 5, powers 0 and 3: m = 1 and n = 125, so log m = 0 and log n = 3. Multiplying by 1 changes nothing, so the product is 125 and its logarithm is 0 + 3 = 3.
  4. Any base b: m = b to the power p and n = b to the power q give mn = b to the power p + q, so log mn = p + q = log m + log n.

The three laws of logarithms (same base b, positive m and n):

logb (mn) = logb m + logb n
logb (m ÷ n) = logb m − logb n
logb (mk) = k logb m

They are the index laws in disguise: multiplying numbers adds powers, dividing subtracts powers, and raising a power to a power multiplies the powers.

Why they work. Let logb m = p and logb n = q, so m = bp and n = bq.

  1. mn = bp × bq = bp + q, so logb (mn) = p + q.
  2. m ÷ n = bp ÷ bq = bp − q, so logb (m ÷ n) = p − q.
  3. mk = (bp)k = bpk, so logb (mk) = pk = k logb m.

Two common slips. Multiplying instead of raising: 3 log2 4 is log2 (43) = log2 64 = 6, not log2 12. And dividing the logarithms: log2 64 ÷ log2 8 = 6 ÷ 3 = 2, but log2 (64 ÷ 8) = 3, so the quotient law subtracts.

Worked examples

1. Write log6 4 + log6 9 as one logarithm and evaluate it.

  1. Adding logarithms multiplies the numbers: log6 (4 × 9) = log6 36.
  2. 36 = 62, so the value is 2.

2. Evaluate log5 250 − log5 2.

  1. Subtracting divides the numbers: log5 (250 ÷ 2) = log5 125.
  2. 125 = 53, so the value is 3.

3. Write 2 loga 3 + loga 5 as a single logarithm.

  1. The power law first: 2 loga 3 = loga 32 = loga 9.
  2. Then the product law: loga 9 + loga 5 = loga 45.

Your go. The first step is given. Evaluate log10 400 + log10 25. The product is 400 × 25 = 10 000.

  1. Combine into one logarithm: log10 10 000.
  2. Write 10 000 as a power of 10 and read off the logarithm.
Check the answer

10 000 = 104, so the value is 4.

Check before you go on

1. Which is equal to log2 (8x), for x > 0?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Evaluate log2 5 + log2 6.4.

3. Evaluate log3 4374 − log3 2.

Level 2 · Building

4. Evaluate 2 log2 48 − log2 2.25.

5. Given that loga 2 = 0.3 and loga 5 = 0.7, find loga 40.

Level 3 · Stretch

6. Solve log3 x + log3 (x − 6) = 3.

7. Solve log5 (x + 1) − log5 x = 1, as a decimal.

Something to think about

The quotient law says logb (m ÷ n) = logb m − logb n. What does it give when m and n are the same number, and does that match the value of logb 1?

One way to think about it

When m = n the quotient is 1 and the right-hand side is logb m − logb m = 0. That matches logb 1 = 0, which is the logarithm statement of b0 = 1, so the quotient law is consistent with the definition.

Exit ticket

8. What is the value of log2 (16 + 16)?