Year 11 Extended · Unit 6 Logarithms and non-linear inequalities · Lesson 1 of 8

How many steps does it take to reach a target?

A quantity that is multiplied by the same factor every step grows faster than any quantity that adds the same amount. This lesson sets up the question that logarithms answer: what power of the base reaches a target?

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 Year 8: work out 34.
    Answer81. 3 × 3 × 3 × 3.
  2. From Year 9: increase 400 by 10%.
    Answer440.
  3. From Year 10: what comes next in 5, 10, 20, 40?
    Answer80. Each term is double the one before.
Words used in this lesson
exponential growth
increase by the same factor in each step. For example: a colony that triples every hour.
factor
the number each step multiplies by, written b. For example: b = 1.5.
start value
the amount at step 0, written a. For example: a = 200 rupees.
decay
decrease by a factor between 0 and 1 in each step. For example: a factor of 0.9.
power
the number of copies of the base that are multiplied together. For example: the 4 in 74 = 7 × 7 × 7 × 7.

How long until the tray is full?

In a laboratory in Pune a colony starts as a single cell, and every hour each cell splits into two. The tray holds about one thousand cells before they begin to crowd each other. Writing out the hours: 1, 2, 4, 8 and so on.

Predict first: roughly how many hours will it take: closer to 5, to 50 or to 500? Then use the figure to see how a base and a target behave, and decide whether the answer has to be a whole number.

3
25
3whole steps to reach the target
27value after those steps

Base 3, target 25: 3 to the power 2 is 9, which is below 25, and 3 to the power 3 is 27, which passes it, so the power is between 2 and 3.

Show me the steps
  1. Base 4, target 40: the powers are 1, 4, 16, 64. The value 16 is short and 64 passes 40, so the power is between 2 and 3 and 3 whole steps are needed.
  2. Base 5, target 25: 5 to the power 2 is exactly 25, so the power is exactly 2 and no other value is needed.
  3. Base 2, target 25: the powers are 1, 2, 4, 8, 16, 32. The value 16 is short and 32 passes 25, so the power is between 4 and 5 and 5 whole steps are needed.
  4. Any base b and target t: find the first power of b that is at least t. If it equals t, the power is a whole number. If it passes t, the power is between that step and the one before.

Exponential growth multiplies by the same factor each step. A start value a and a factor b give y = a × bx after x steps. A linear model adds the same amount each step; an exponential model multiplies by the same amount, so a list of values has a constant ratio instead of a constant difference.

"What power?" asks for x when a × bx has reached a target. Count powers of the base until one reaches the target: if it lands exactly the answer is a whole number, and if it jumps past the target the answer lies between two whole numbers.

Two common slips. Reading 25 as 2 × 5: the power counts how many 2s are multiplied, so 25 = 2 × 2 × 2 × 2 × 2, not 10. And using the target's size as the answer: if 10n must reach 700, then n is the number of steps, a small number, not 700 and not 70.

Worked examples

1. A savings pot of 200 rupees is multiplied by 1.5 each year. How much is there after 3 years?

  1. The model is 200 × 1.5x. Three steps: 1.5 × 1.5 = 2.25, and 2.25 × 1.5 = 3.375.
  2. 200 × 3.375 = 675 rupees.

2. What power of 3 gives 243?

  1. List the powers: 3, 9, 27, 81, 243. The value 243 is the fifth in the list, and the first power is 31.
  2. So 35 = 243 and the power is 5.

3. A plant of height 5 cm triples in height every day. On which day does it first pass 1000 cm?

  1. Height after n days is 5 × 3n. On day 4 it is 5 × 81 = 405, which is below 1000.
  2. On day 5 it is 5 × 243 = 1215, which is above 1000. The first day is day 5.

Your go. The first step is given. A colony starts at 8 cells and doubles every hour. After how many whole hours does it first pass 500 cells? Its size is 8 × 2n.

  1. Find the hour at which 8 × 2n first reaches 500. Divide: 2n must reach 500 ÷ 8 = 62.5.
  2. List the powers of 2 and find the first that is at least 62.5.
Check the answer

25 = 32 is below 62.5 and 26 = 64 is above it, so the answer is 6 hours. Check: 8 × 64 = 512, and 8 × 32 = 256 is not yet enough.

Check before you go on

1. Which list is growing by the same factor each step, rather than by the same amount?

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. A club in Cebu has 12 members and doubles in size every year. How many members will it have after 5 years?

3. Tea in a flask in Nairobi is 80 degrees hotter than the room. The extra heat halves every 10 minutes. How many degrees hotter than the room is the tea after 30 minutes?

Level 2 · Building

4. A rumour in a school in Accra is known by 6 people at 8 am. Every hour the number who know it triples. After how many whole hours will more than 1500 people know it?

5. A quantity starts at 3 and is multiplied by the same factor every step. After 4 steps it is 48. What is the factor?

Level 3 · Stretch

6. A culture of 300 bacteria triples every 2 hours. After how many hours are there 24 300 bacteria?

7. Plan A has 100 rupees and adds 30 rupees every year. Plan B has 10 rupees and doubles every year. After how many whole years does Plan B first have more than Plan A?

Something to think about

A factor smaller than 1 shrinks a quantity: 80, then 40, then 20, and so on. Does the quantity ever reach exactly zero?

One way to think about it

No. Each step takes away half of what is left, and half of anything above zero is still above zero. The amounts get as small as you like but are never zero, which is why the tea in a flask is never exactly the room's temperature.

Exit ticket

8. A quantity is 5 now and is multiplied by 4 at every step. Which formula gives the quantity after x steps?