Year 11 Extended · Unit 6 Logarithms and non-linear inequalities · Lesson 2 of 8

What is a logarithm, and how can one be worked out by hand?

A logarithm is the power a base must be raised to in order to make a number. This lesson reads each logarithm back into a power and evaluates 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 1: what power of 2 gives 64?
    Answer6. 2, 4, 8, 16, 32, 64.
  2. From Year 10: write 1 ÷ 8 as a power of 2.
    Answer2−3.
  3. From Year 10: work out 491⁄2.
    Answer7. The square root of 49.
Words used in this lesson
logarithm
the power a base is raised to in order to make a number. For example: log base 6 of 36.
base
the number that is multiplied by itself in a power. For example: the 6 in 63.
inverse
the operation that undoes another one. For example: taking a square root undoes squaring.
evaluate
to work out the value of an expression. For example: log base 7 of 7.
reciprocal
one divided by a number. For example: the reciprocal of 10 is 0.1.

What is the power of ten?

A warehouse system in Lagos stores big numbers by their size: instead of writing out one hundred thousand, it records how many tens were multiplied together to make it. For one thousand the record is 3, because 10 × 10 × 10 = 1000.

Predict first: what would the record be for one hundred thousand, and what do you think it should be for 0.001? Then use the figure to see the pair of facts, a power forwards and the record backwards.

3
2
9the number
2log of the number to this base

Base 3, power 2: 3 to the power 2 is 9, so the logarithm to base 3 of 9 is 2.

Show me the steps
  1. Base 3, power 2: 3 to the power 2 is 9, so the logarithm to base 3 of 9 is 2.
  2. Base 5, power 0: 5 to the power 0 is 1, so the logarithm to base 5 of 1 is 0, and it is 0 for every base.
  3. Base 4, power 3: 4 to the power 3 is 64, so the logarithm to base 4 of 64 is 3.
  4. Any base b and power x: b to the power x is the number N, and the logarithm to base b of N is x. The two facts say the same thing.

A logarithm is the power. The statement logb N = x means exactly bx = N: the logarithm to base b of N is the power that b must be raised to in order to make N. Because the two statements say the same thing, a logarithm is worked out by asking “what power?”

Three values are the same for every base: logb 1 = 0 (anything to the power 0 is 1), logb b = 1, and logb bk = k. A number below 1 has a negative logarithm when b is above 1, since b−k = 1 ÷ bk.

The base must be positive and not 1, and N must be positive: only positive numbers have a logarithm.

Two common slips. Dividing instead of finding the power: log5 25 is not 25 ÷ 5. It asks how many 5s multiply to 25. And swapping the base and the number: log2 8 = 3, but log8 2 is one third.

Worked examples

1. Evaluate log3 81.

  1. Ask: what power of 3 is 81? List 3, 9, 27, 81.
  2. 81 is the fourth in the list, so 34 = 81 and log3 81 = 4.

2. Evaluate log2 (1 ÷ 16).

  1. 16 = 24, so 1 ÷ 16 = 2−4.
  2. The power is −4.

3. Evaluate log64 4.

  1. The question is: what power of 64 is 4? Since the cube root of 64 is 4, 641⁄3 = 4.
  2. log64 4 = one third, 0.3333 to four decimal places.

4. Find x if logx 81 = 4.

  1. Read it as a power: x4 = 81.
  2. Take the fourth root: x = 3.

Your go. The first step is given. Evaluate log5 (1 ÷ 125). Write 125 as a power of 5: 125 = 5 × 5 × 5.

  1. 125 is a power of 5, so 1 ÷ 125 is a negative power of 5.
  2. Use b−k = 1 ÷ bk to read off the power.
Check the answer

125 = 53, so 1 ÷ 125 = 5−3 and the logarithm is −3.

Check before you go on

1. Which statement says the same thing as 27 = 128?

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 32.

3. Evaluate log10 0.01.

Level 2 · Building

4. Find x if logx 1024 = 5.

5. Evaluate log36 6, as a decimal.

Level 3 · Stretch

6. Find x if log4 x = −2, as a decimal.

7. Find b if logb 343 = 1.5.

Something to think about

The statement 2x = 0 asks for a power of 2 that gives zero. What happens as x gets more and more negative, and what does that say about log2 0?

One way to think about it

As x gets more negative, 2x gets smaller and smaller (a half, a quarter, an eighth) but never reaches zero. No power of 2 gives zero, so log2 0 does not exist.

Exit ticket

8. Which statement is true?