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

What does one step on a scale of readings really mean?

Acidity, earthquake size and loudness are each reported as one short reading. This lesson asks what a step on such a scale means for the amount behind it, and turns a difference in readings into a comparison of amounts.

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 2: evaluate log10 1000.
    Answer3.
  2. From Year 9: write 106 as an ordinary number.
    Answer1 000 000.
  3. From lesson 5: find log10 2 to two decimal places.
    Answer0.30.
Words used in this lesson
logarithmic scale
a scale whose readings are the logarithms of the amounts. For example: pH or decibels.
pH
a scale of acidity, lower for stronger acids. For example: a lemon is acidic.
magnitude
the number used to report the size of an earthquake. For example: a magnitude 5.9 earthquake.
decibel
a unit of sound level, ten times the logarithm of a ratio of intensities. For example: dB.
intensity
the power of a sound per unit area. For example: W/m2.

How much bigger is a magnitude 8 than a magnitude 6?

Two earthquakes hit different regions of Chile in the same year: one measured 6 on the magnitude scale and one measured 8. A news report says the second was “a third bigger”, because 8 is a third more than 6.

Predict first: is the report right about how much more the ground moved, and if not, roughly how many times more? Then use the figure to compare two readings on a straight line of amounts and on a logarithmic line.

2
3
1difference in readings
10times as large (larger amount over smaller)

Reading A is 2 and reading B is 3: the difference is 1, so B is 10 times as large as A.

Show me the steps
  1. Readings 2 and 3: the amounts are 100 and 1000, and the difference of 1 means one factor of 10, so B is 10 times as large as A.
  2. Readings 0 and 5: the amounts are 1 and 100 000, and the difference of 5 means five factors of 10, so B is 100 000 times as large as A.
  3. Readings 3 and 2: the amounts are 1000 and 100, and the difference of 1 means one factor of 10, so A is 10 times as large as B. The larger reading is the larger amount whichever slider holds it.
  4. Any two readings differing by d: the larger amount is 10 to the power d times the smaller, whatever the readings themselves are.

A logarithmic scale reports the exponent. A reading R stands for an amount of 10R units, so R = log10 (amount). Each step of 1 multiplies the amount by 10, and a difference of d in two readings means the larger amount is 10d times the smaller.

pH measures acidity as the negative of the logarithm of the concentration, so a lower pH is more acidic. Earthquake magnitude adds 1 for each factor of 10 in the size of the ground movement. Decibels give the level of a sound as L = 10 log10 (I ÷ I0), so each 10 dB is a factor of 10 in intensity.

Two common slips. Comparing readings by dividing or subtracting as if they were amounts: 90 dB is not three times 30 dB, because the gap of 60 dB is a factor of 1 000 000 in intensity. And mixing the scale's direction: a lower pH means more acid, so the smaller reading is the larger amount of acid.

Worked examples

1. Lemon juice has pH 2 and soap has pH 7. How many times more acidic is the lemon juice?

  1. The difference is 7 − 2 = 5 steps, and each step is a factor of 10.
  2. Lemon juice is 105 = 100 000 times more acidic.

2. An earthquake of magnitude 7.4 follows one of magnitude 5.9. How many times larger is its amplitude?

  1. The difference is 7.4 − 5.9 = 1.5, so the ratio is 101.5.
  2. With the 10x key, 101.5 = 31.6, so about 32 times.

3. A sound has intensity 4 × 10−4 W/m2 and I0 = 10−12 W/m2. Find its level.

  1. I ÷ I0 = 4 × 108, and log (4 × 108) = 8.602.
  2. L = 10 × 8.602 = 86.0 dB.

Your go. The first step is given. A sound has intensity 10−6 W/m2. Use I0 = 10−12 W/m2 and find L. The ratio is 10−6 ÷ 10−12.

  1. Divide the powers of 10 by subtracting the exponents to find the ratio.
  2. Take the logarithm of the ratio and multiply by 10.
Check the answer

The ratio is 106, its logarithm is 6, and L = 10 × 6 = 60 dB.

With a calculator

For a power of 10, use the key marked 10x, which is often the second function of the log key. To find 103.5, press the second function key, then the log key, then type 3.5 and equals. The display should read between 1000 and ten thousand, because 3.5 lies between 3 and 4.

For a decibel level, type 10 × log, then the ratio in brackets. To find 10 log 5000, enter 10 × log (5000) and read 37.0. Work out the ratio of intensities completely before you take the logarithm, and write the 10 as a multiplier outside it. Round only at the end.

Check before you go on

1. Rain water is 1000 times as acidic as sea water of pH 8. What is the pH of the rain water?

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 lake in Kenya has pH 9 and a river nearby has pH 7. How many times more acidic is the river?

3. A road drill is at 70 dB and a conversation is at 40 dB. How many times greater is the intensity of the drill?

Level 2 · Building

4. An earthquake in Nepal has an amplitude 1000 times that of an earthquake of magnitude 3.5. What is its magnitude?

5. A sound has an intensity 105 times the threshold intensity I0. Using L = 10 log10 (I ÷ I0), find its level in decibels.

Level 3 · Stretch

6. A sound has intensity 3.2 × 10−5 W/m2. Using I0 = 10−12 W/m2, find its level L = 10 log10 (I ÷ I0) in decibels, to one decimal place.

7. One machine in a workshop in Manila produces 80 dB. A second identical machine starts beside it. What is the combined level, to the nearest whole decibel?

Something to think about

Each step of the magnitude scale multiplies the ground movement by a fixed factor. Would a scale that just counted the movement, in whole units, be better for warning the public, or worse?

One way to think about it

There is no single right answer. A plain count would match how many people think, but it would need enormous numbers: the strongest quakes move the ground millions of times more than the weakest felt ones. A logarithmic scale keeps the readings short at the cost of making each step hard to judge.

Exit ticket

8. Which pair of earthquake magnitudes has amplitudes in the ratio 10 000 to 1?