Year 11 Extended · Unit 6 Logarithms and non-linear inequalities · Lesson 4 of 8

How can an unknown power be found exactly?

An unknown in a power can be found by matching bases when both sides are powers of one number, and by writing it as a logarithm when they are not. This lesson does both exactly and brackets each answer between whole numbers.

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 log5 125.
    Answer3.
  2. From Year 9: write 16 as a power of 2.
    Answer24.
  3. From Year 10: simplify (23)x.
    Answer23x.
Words used in this lesson
exponent
the power a number is raised to. For example: the 4 in 54.
isolate
to get a term alone on one side of an equation. For example: divide by 6 to get 3x alone.
match the bases
to write both sides as powers of the same number. For example: 9 as 32.
exact value
an answer left as a logarithm or root, not rounded. For example: log base 7 of 30.
reject
to throw out a root that does not fit the problem. For example: u = −5 for u = 3x.

When will the hospital run out of beds?

A new virus appears in a district of Bangladesh with 4 reported cases. Every week the number of cases triples. The district hospital has 500 beds.

Predict first: roughly in which week will the number of cases first pass 500, and is the exact moment at the end of a whole week or part way through one? Then use the figure to see why an equation like this has a solution even when the target is not a power of the base.

2
20
4.32solution x (2 decimal places)
4whole number just below it

Base 2, target 20: 2 to the power 4 is 16 and 2 to the power 5 is 32, so x lies between 4 and 5. It is log base 2 of 20, which is about 4.32.

Show me the steps
  1. Base 2, target 20: 2 to the power 4 is 16 and 2 to the power 5 is 32, so x lies between 4 and 5. It is log base 2 of 20, about 4.32.
  2. Base 2, target 32: 2 to the power 5 is exactly 32, so the curve crosses the line at x = 5, a whole number.
  3. Base 4, target 10: 4 to the power 1 is 4 and 4 to the power 2 is 16, so x lies between 1 and 2. It is log base 4 of 10, about 1.66.
  4. Any base b and target N: the crossing is at x = log base b of N. If N is a power of b the crossing is at a whole number; otherwise it lies between the two whole numbers whose powers surround N.

To solve bx = N there are two routes.

Same base. If N is a power of b, write both sides as powers of one base. Equal bases mean equal powers, so the exponents can be set equal: 23x = 25 gives 3x = 5.

Logarithm. If N is not a power of b, the solution is x = logb N, which is exact. List the powers of b either side of N to say between which whole numbers it lies.

If the unknown power is multiplied by a number, divide first: in 3 × 2x = 48 the power 2x must be isolated before it is read.

Two common slips. Dividing instead of finding the power: in 3x = 20, x is not 20 ÷ 3, which is about 6.7 (36.7 is far more than 20). And multiplying the bases: 2 × 3x = 54 is not 6x = 54. Divide by 2 first to get 3x = 27.

Worked examples

1. Solve 8x = 32.

  1. Both are powers of 2: 8 = 23 and 32 = 25, so 23x = 25.
  2. Equal bases, so 3x = 5 and x = 5⁄3.

2. Solve 9x − 1 = 3x + 4.

  1. Write 9 as 32: 32(x − 1) = 3x + 4, so 2x − 2 = x + 4.
  2. Subtract x and add 2: x = 6.

3. Solve 6 × 3x = 162.

  1. Divide by 6 to isolate the power: 3x = 27.
  2. 27 = 33, so x = 3.

4. Solve 7x = 30 exactly, and say where it lies.

  1. 30 is not a power of 7, so x = log7 30.
  2. 71 = 7 and 72 = 49, so x lies between 1 and 2.

Your go. The first step is given. Solve 6x = 100 exactly and say between which two whole numbers x lies. Powers of 6: 62 = 36 and 63 = 216.

  1. 100 is not a power of 6, so write the solution as a logarithm.
  2. Compare 100 with 36 and 216 to bracket it.
Check the answer

x = log6 100, and since 36 < 100 < 216 it lies between 2 and 3.

Check before you go on

1. Which statement is true for the solution of 5x = 300?

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. Solve 9x = 27, as a decimal.

3. Solve 4 × 2x = 128.

Level 2 · Building

4. Solve 8x − 1 = 4x + 1.

5. The solution of 3x = 100 lies between two consecutive whole numbers. What is the smaller one?

Level 3 · Stretch

6. Solve 32x − 4 × 3x − 45 = 0.

7. The amount of a medicine in the body is N = 640 × 0.5t ÷ 3 milligrams after t hours. After how many hours is N equal to 10?

Something to think about

In 3x = 200 the answer is log3 200, which lies between 4 and 5. Is it closer to 4 or to 5, and is it as far along as 200 is between 81 and 243?

One way to think about it

Closer to 5. 200 is 119 above 81 but only 43 below 243. Sharing the gap in a straight line would put x about 0.73 of the way from 4 to 5 (119 ÷ 162), but the curve bends upwards, so the crossing is further along: log3 200 is about 4.82.

Exit ticket

8. Which of these equations has the solution x = 4?