Year 11 Extended · Unit 6 Logarithms and non-linear inequalities · Lesson 5 of 8

How can a calculator find a logarithm to any base?

A calculator has a logarithm key for base 10, and the change of base law turns it into a logarithm to any base. This lesson derives the law, uses it to put decimal values on exponential equations, and solves growth problems in whole years.

Number and AlgebraScientific 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 3: write log2 12 + log2 5 as one logarithm.
    Answerlog2 60.
  2. From lesson 4: solve 2x = 128.
    Answerx = 7.
  3. From Year 10: increase 250 by 4%.
    Answer260.
Words used in this lesson
change of base
writing a logarithm as a quotient of two logarithms with a new base. For example: log 9 ÷ log 4.
natural logarithm
the logarithm to the base called e, written ln. For example: the ln key.
compound growth
growth applied to the amount at the end of each period. For example: 5% a year on the new total.
round up
to go to the next whole number even if the first decimal is small. For example: 11.9 years becomes 12 years.
exceed
to be greater than. For example: more than 20 000 people.

How long does it take to double?

A saver in Colombo puts money in an account that adds 5% a year. After t years each rupee has become 1.05t rupees. She wants to know how many years it takes for the money to double, so 1.05t = 2. The number 2 is not a power of 1.05 that you can list.

Predict first: roughly how many years will it take for the money to double, and would a rate twice as big halve the time? Then use the figure to see how a logarithm to any base can be counted in copies of another logarithm.

3
50
3.56log base b of N (2 decimal places)
3whole copies that fit

Base 3, number 50: log 50 is 1.7 and log 3 is 0.48, so 3 whole copies of log 3 fit and part of another. The logarithm to base 3 of 50 is 3.56.

Show me the steps
  1. Base 3, number 50: log 50 is about 1.70 and log 3 is about 0.48, so 3 whole copies fit and a part copy, 3.56 copies in all.
  2. Base 9, number 81: log 81 is about 1.91 and log 9 is about 0.95, so exactly 2 copies fit and the logarithm is 2.
  3. Base 2, number 10: log 10 is 1 and log 2 is about 0.30, so 3 whole copies fit and a part copy, 3.32 copies in all.
  4. Any base b and number N: the logarithm to base b of N is log N divided by log b, the number of copies of log b that fit into log N.

Change of base. A logarithm to any base can be found from logarithms to a base c that you have:

logb N = logc N ÷ logc b.

With a calculator c is 10, so logb N = log N ÷ log b. It follows that bx = N has solution x = log N ÷ log b, and that swapping the base and the number takes the reciprocal: logb a = 1 ÷ loga b.

Why: if x = logb N then bx = N. Take logc of both sides and use the power law: x logc b = logc N, so x = logc N ÷ logc b.

Two common slips. Dividing inside the logarithm: log N ÷ log b is not log (N ÷ b). For example log 100 ÷ log 10 = 2 ÷ 1 = 2, but log (100 ÷ 10) = log 10 = 1. And putting the base on top: log8 30 asks about the number 30, so log 30 is the top and log 8 is underneath. Swapped, it gives the reciprocal, log30 8.

Worked examples

1. Find log5 80 to two decimal places.

  1. log5 80 = log 80 ÷ log 5 = 1.9031 ÷ 0.6990.
  2. The division gives 2.7227, so 2.72. Check: 52 = 25 and 53 = 125 surround 80.

2. In how many years does money double at 6% a year? Solve 1.06t = 2.

  1. t = log 2 ÷ log 1.06 = 0.30103 ÷ 0.025306.
  2. The division gives 11.896, so about 11.9 years.

3. Solve 3 × 7x = 500.

  1. Isolate the power: 7x = 500 ÷ 3 = 166.67.
  2. x = log 166.67 ÷ log 7 = 2.2218 ÷ 0.8451 = 2.63.

Your go. The first step is given. Solve 4x = 90 to two decimal places. Use x = log 90 ÷ log 4.

  1. Type the logarithm of the number first, then divide by the logarithm of the base.
  2. Check the size: 43 = 64 and 44 = 256.
Check the answer

1.9542 ÷ 0.6021 = 3.2459, so x = 3.25, between 3 and 4 as expected.

With a calculator

Use the key marked log, which is base 10. To find log4 200, type log 200, then ÷, then log 4, then equals. Close the bracket if the calculator opens one: some models open a bracket when the log key is pressed. Do not type the base as a subscript.

Estimate first. 44 is 256, which is just above 200, so the answer should be a little under 4 (it is 3.82). If the display is close to that, the keys were right. Keep every digit in the calculator and round only the final answer. Many calculators have a second key marked ln, a different base; either key works if you use the same one on the top and the bottom.

Check before you go on

1. Which single logarithm is equal to log2 7 ÷ log2 5?

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. Use a calculator to find log7 300 to two decimal places.

3. Solve 5x = 700 to two decimal places.

Level 2 · Building

4. A town in Ecuador has 12 000 people and grows by 3% a year. After how many whole years will the population first exceed 20 000?

5. A fund grows by 7% a year. After how many whole years will it first be more than three times its starting value?

Level 3 · Stretch

6. Solve 2x = 3x − 1 to two decimal places.

7. Solve 32x + 1 = 4x to two decimal places.

Something to think about

Without a calculator, find log2 3 × log3 4 × log4 8. What would the longer chain log2 3 × log3 4 × … × log15 16 give?

One way to think about it

Write each as a quotient of base 10 logarithms: (log 3 ÷ log 2) × (log 4 ÷ log 3) × (log 8 ÷ log 4). Each top cancels the next bottom, leaving log 8 ÷ log 2 = log2 8 = 3. The longer chain leaves log 16 ÷ log 2 = log2 16 = 4.

Exit ticket

8. Which value is closest to log3 400?