Chapter 3 of 7

All 7 chapters
  1. 1Exponential functions, readunderstood
  2. 2Logarithms, readunderstood
  3. 3Laws of logarithms, readunderstood
  4. 4Equations with powers, readunderstood
  5. 5The number e and natural logs, readunderstood
  6. 6Growth and decay models, readunderstood
  7. 7Straight lines from logs, readunderstood

M5.2.3 Chapter 3

Laws of logarithms

Logarithms are powers, so they should follow the laws of powers. What do they let us do with a product, a quotient and a power?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use the laws for the log of a product, quotient and power
  • Write 2 log 3 + log 4 as one logarithm

The laws of indices say a³ × a⁴ = a⁷: to multiply powers of one base, add the powers. A logarithm is a power. So adding logarithms should match multiplying numbers. That is the first law, the product law, and the picture shows it with lengths.

Two rows of bars drawn to scale. The top row has a hatched bar named log 2, with value 0.301, then a dotted bar named log 3, with value 0.477, end to end. The bottom row is one bar named log 6, with value 0.778, as long as the two above together. Dashed lines join the start and the end of the two rows. The line under them reads log 2 + log 3 = log 6.log 2log 3log 60.3010.4770.778log 2 + log 3 = log 6
Lengths drawn to scale, in base 10. The bar for log 2 and the bar for log 3, end to end, are as long as the bar for log 6.

Here are the three laws. They hold for logs to any one base a, with a > 0 and a ≠ 1, as long as every number inside a log is above 0. The base must be the same throughout.

A table with the head The three laws. Rows: log xy = log x + log y; log (x/y) = log x − log y; log (xⁿ) = n log x.The three lawslog xy = log x + log ylog (x/y) = log x − log ylog (xⁿ) = n log x
The product law, the quotient law and the power law, for x above 0 and y above 0. The power n can be any real number.

Worked example

Prove the product law: for x > 0 and y > 0, log (xy) = log x + log y, with every log to the base a.

  1. Let p be the log of x to the base a, and q the log of y. By the definition, x = a to the power p and y = a to the power q.
  2. Multiply: xy = (a to the power p) × (a to the power q).
  3. The law of indices says to add the powers: xy = a to the power (p + q).
  4. Read that as a logarithm: the power of a that gives xy is p + q. So the log of xy is p + q.
  5. Put back p and q: log (xy) = log x + log y.

Answer: log (xy) = log x + log y.

The other two laws come the same way. For the quotient law, x ÷ y is (a to the power p) ÷ (a to the power q), which is a to the power (p − q). So the log of x ÷ y is p − q. For the power law, xⁿ is (a to the power p) raised to n, which is a to the power np. So the log of xⁿ is np. Two special cases are worth knowing: n = −1 gives log (1/x) = −log x, and n = 1/2 gives log √x = 1/2 log x.

Two rows of bars drawn to scale. The top row has a hatched bar named log 3, with value 0.477, then a dotted bar named log 4, with value 0.602, end to end. The bottom row is one bar named log 12, with value 1.079, as long as the two above together. Dashed lines join the start and the end of the two rows. The line under them reads log 3 + log 4 = log 12.log 3log 4log 120.4770.6021.079log 3 + log 4 = log 12
Take the bar for log 3 away from the bar for log 12 and the bar for log 4 is left. That is the quotient law: log 12 − log 3 = log 4.

Worked example

Write 2 log 3 + log 4 as one logarithm.

  1. Use the power law on the first term: 2 log 3 = log (3²) = log 9.
  2. The sum is now log 9 + log 4.
  3. Use the product law and multiply the numbers inside: log 9 + log 4 = log (9 × 4) = log 36.
  4. So 2 log 3 + log 4 can be written as one logarithm, log 36.

Answer: 2 log 3 + log 4 = log 36.

Worked example

Given that log 2 = a and log 3 = b, with every log to base 10, write each of these in terms of a and b. (i) log 72. (ii) log 1.5. (iii) log 5.

  1. (i) 72 = 2³ × 3². So log 72 = log 2³ + log 3² = 3 log 2 + 2 log 3 = 3a + 2b.
  2. (ii) 1.5 = 3 ÷ 2. So log 1.5 = log 3 − log 2 = b − a.
  3. For (iii) we need 5 in terms of 2. Use 5 = 10 ÷ 2.
  4. In base 10 the log of 10 is 1. Since 10¹ = 10, the log of 10 is 1.
  5. (iii) log 5 = log 10 − log 2 = 1 − a.

Answer: (i) 3a + 2b, (ii) b − a, (iii) 1 − a.

Worked example

Show that log₂ 48 − log₂ 3 = 4.

  1. Both logs are to the base 2, so the quotient law applies.
  2. log₂ 48 − log₂ 3 = log₂ (48 ÷ 3).
  3. Work out the division: 48 ÷ 3 = 16.
  4. Since 2⁴ = 16, log₂ 16 = 4.
  5. So log₂ 48 − log₂ 3 = 4, as required.

Answer: log₂ 48 − log₂ 3 = log₂ 16 = 4.

There is one more rule, for a calculator that has only a base 10 key and a natural log key. The change of base formula lets you change the base of any logarithm. Here is the proof, then an example.

Worked example

Prove that the log of x to the base a is log x ÷ log a. The logs on the right are to any one base.

  1. Let y be the log of x to the base a. By the definition, a to the power y is x.
  2. Take the log of both sides, to any base: log (a to the power y) = log x.
  3. The power law moves y to the front: y log a = log x.
  4. Divide both sides by log a. This is allowed, because a ≠ 1 means log a is not 0: y = log x ÷ log a.
  5. So the log of x to the base a is log x ÷ log a.

Answer: The log of x to the base a is log x ÷ log a.

Worked example

Find log₅ 40 to 3 significant figures.

  1. The calculator log key is base 10, so change the base: log₅ 40 = log 40 ÷ log 5.
  2. log 40 = 1.60206 and log 5 = 0.69897, each to 5 decimal places.
  3. Divide: 1.60206 ÷ 0.69897 = 2.292.
  4. To 3 significant figures, log₅ 40 = 2.29.
  5. Check: 5² = 25 and 5³ = 125, and 40 is between them. So the answer must be between 2 and 3, and it is.

Answer: 2.29

See it three ways

With things

Two sticks are cut. One is log 4 units long, which is 0.602 units. The other is log 5 units long, which is 0.699 units. Laid end to end they are 1.301 units long, which is log 20.

As a picture

Two rows of bars drawn to scale. The top row has a hatched bar named log 4, with value 0.602, then a dotted bar named log 5, with value 0.699, end to end. The bottom row is one bar named log 20, with value 1.301, as long as the two above together. Dashed lines join the start and the end of the two rows. The line under them reads log 4 + log 5 = log 20.log 4log 5log 200.6020.6991.301log 4 + log 5 = log 20

In words

Adding the lengths of two logs gives the length of the log of the product. Multiplying numbers has become adding lengths.

In symbols

log 4 + log 5 = log (4 × 5) = log 20, and 0.602 + 0.699 = 1.301.

Try it

Test the product law

You need: a scientific calculator, paper and a pencil.

  1. Find log 8 and log 5. Write each to 3 decimal places.
  2. Add the two results.
  3. Now find log 40, which is 8 × 5, to 3 decimal places. Compare it with the sum.

What you will see: The sum of log 8 and log 5 and the value of log 40 are the same, 1.602. Adding logs gives the log of the product. Now try log 8 × log 5, and you get a different number.

If you cannot try it, this is what happens.
A table with the heads expression and value. Rows: log 8 and 0.903; log 5 and 0.699; log 8 + log 5 and 1.602; log 40 and 1.602.expressionvaluelog 80.903log 50.699log 8 + log 51.602log 401.602

A wrong idea many people have

log (a + b) = log a + log b.

Why it feels right: It looks like the way brackets spread in 2(a + b) = 2a + 2b. The log seems to be handed out to each term.

The test that breaks it: Put a = 10 and b = 10 in base 10. Then log (10 + 10) = log 20 = 1.301. But log 10 + log 10 = 1 + 1 = 2. The two sides are not equal.

What is true: A log is not a number that multiplies a bracket. It is a function, and log (a + b) cannot be split. The law adds logs only for a product: log a + log b = log (ab). Catch the slip by asking what the sign inside the bracket is. Only a product or a quotient inside lets a law work.

A wrong idea many people have

log x ÷ log y = log (x ÷ y), or log x − log y = log x ÷ log y.

Why it feels right: The quotient law has a division in it, so it is easy to put the division in the wrong place.

The test that breaks it: In base 10 take x = 100 and y = 10. Then log x ÷ log y = 2 ÷ 1 = 2. But log (x ÷ y) = log 10 = 1, and log x − log y = 2 − 1 = 1.

What is true: The difference of two logs is the log of the quotient: log x − log y = log (x ÷ y). The quotient of two logs, log x ÷ log y, has no simpler form. It appears only in the change of base formula.

A wrong idea many people have

(log x)² = 2 log x, and log (xy) = (log x)(log y).

Why it feels right: The power law moves a power to the front, and the product law has two logs in it. The pieces look interchangeable.

The test that breaks it: In base 10 take x = 10. Then (log x)² = 1² = 1, but 2 log x = 2. Also take x = 10 and y = 100: log (xy) = log 1 000 = 3, but (log x)(log y) = 1 × 2 = 2.

What is true: The power law is about a power inside the log: log (x²) = 2 log x. The square of a log, (log x)², is a different number. And the log of a product is the sum of the logs, not their product.

Key words

product law
log x + log y = log (xy), for logs to the same base and x, y above 0.
quotient law
log x − log y = log (x ÷ y).
power law
n log x = log (xⁿ), where n is any real number.
change of base formula
the log of x to the base a is log x ÷ log a, with both logs to any one base.

Back to your guess

The answer is A: log 10 + log 5 = log 50.

Adding logs multiplies the numbers inside. In base 10, log 10 = 1 and log 5 = 0.699, so the sum is 1.699. And log 50 = 1.699, because 50 = 10 × 5. The sum is not log 15. This chapter shows why, and what else follows.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which of these is equal to 2 log 5?

    Tap your answer.

    1. Yes, that is right. The power law moves the 2 to become a power inside the log: 2 log 5 = log (5²) = log 25.

    2. Not this one. You multiplied 2 by 5 and put the result inside the log. The power law makes the 2 a power: 2 log 5 = log (5²) = log 25. Go to M5.2.3. Try another.

    3. Not this one. That is the square of the number log 5. But 2 log 5 is twice that number. The power law turns 2 log 5 into the log of a square, log (5²). Go to M5.2.3. Try another.

    4. Not this one. You added 2 and 5 and put the result inside the log. The 2 is a multiplier, so it becomes a power: 2 log 5 = log 25. Go to M5.2.3. Try another.

    See the answer

    log 25

    The power law moves the 2 to become a power inside the log: 2 log 5 = log (5²) = log 25.

    • If you chose B: You multiplied 2 by 5 and put the result inside the log. The power law makes the 2 a power: 2 log 5 = log (5²) = log 25. Go to M5.2.3.
    • If you chose C: That is the square of the number log 5. But 2 log 5 is twice that number. The power law turns 2 log 5 into the log of a square, log (5²). Go to M5.2.3.
    • If you chose D: You added 2 and 5 and put the result inside the log. The 2 is a multiplier, so it becomes a power: 2 log 5 = log 25. Go to M5.2.3.

  2. Which statement is true for all x > 0 and y > 0, with every log to the same base?

    Tap your answer.

    1. Not this one. Put x = 1 and y = 1. The left side is log 2, which is not 0. The right side is 0 + 0 = 0. Go to M5.2.3. Try another.

    2. Not this one. Put x = 100 and y = 10 in base 10. The left side is log 90. The right side is 2 ÷ 1 = 2. They are different, and x − y can even be negative. Go to M5.2.3. Try another.

    3. Yes, that is right. This is the product law. The other three are common mistakes. A sum inside a log cannot be split. A quotient inside a log gives a difference of logs, not a quotient of logs.

    4. Not this one. Put x = 100 and y = 10 in base 10. The left side is log 10 = 1. The right side is 2 ÷ 1 = 2. The quotient law has a minus sign: log (x ÷ y) = log x − log y. Go to M5.2.3. Try another.

    See the answer

    log (xy) = log x + log y

    This is the product law. The other three are common mistakes. A sum inside a log cannot be split. A quotient inside a log gives a difference of logs, not a quotient of logs.

    • If you chose A: Put x = 1 and y = 1. The left side is log 2, which is not 0. The right side is 0 + 0 = 0. Go to M5.2.3.
    • If you chose B: Put x = 100 and y = 10 in base 10. The left side is log 90. The right side is 2 ÷ 1 = 2. They are different, and x − y can even be negative. Go to M5.2.3.
    • If you chose D: Put x = 100 and y = 10 in base 10. The left side is log 10 = 1. The right side is 2 ÷ 1 = 2. The quotient law has a minus sign: log (x ÷ y) = log x − log y. Go to M5.2.3.

  3. Work out log₃ 6 + log₃ 1.5.

    Tap your answer.

    1. Yes, that is right. The product law gives log₃ 6 + log₃ 1.5 = log₃ (6 × 1.5) = log₃ 9. Since 3² = 9, this is 2.

    2. Not this one. You added the numbers 6 and 1.5. The sum of two logs is the log of the product, 6 × 1.5. Go to M5.2.3. Try another.

    3. Not this one. You worked out the product 6 × 1.5 = 9 but then stopped. The sum is log₃ 9, and log₃ 9 is the power of 3 that gives 9. Go to M5.2.3. Try another.

    4. Not this one. You divided 6 by 1.5. A sum of logs goes with a product, not a quotient. Go to M5.2.3. Try another.

    See the answer

    2

    The product law gives log₃ 6 + log₃ 1.5 = log₃ (6 × 1.5) = log₃ 9. Since 3² = 9, this is 2.

    • If you chose B: You added the numbers 6 and 1.5. The sum of two logs is the log of the product, 6 × 1.5. Go to M5.2.3.
    • If you chose C: You worked out the product 6 × 1.5 = 9 but then stopped. The sum is log₃ 9, and log₃ 9 is the power of 3 that gives 9. Go to M5.2.3.
    • If you chose D: You divided 6 by 1.5. A sum of logs goes with a product, not a quotient. Go to M5.2.3.

  4. Which of these is equal to log √7?

    Tap your answer.

    1. Not this one. That is log (7²) = log 49. A square root is the power 1/2, not 2. Go to M5.2.3. Try another.

    2. Not this one. That is the square of the number log 7. A square root inside the log is a power inside the log, so the power law moves it to the front. Go to M5.2.3. Try another.

    3. Not this one. That is log (7 ÷ 2). A square root is not a division by 2. It is the power 1/2. Go to M5.2.3. Try another.

    4. Yes, that is right. The square root of 7 is 7 to the power 1/2. The power law brings that power to the front: log √7 = 1/2 log 7.

    See the answer

    1/2 log 7

    The square root of 7 is 7 to the power 1/2. The power law brings that power to the front: log √7 = 1/2 log 7.

    • If you chose A: That is log (7²) = log 49. A square root is the power 1/2, not 2. Go to M5.2.3.
    • If you chose B: That is the square of the number log 7. A square root inside the log is a power inside the log, so the power law moves it to the front. Go to M5.2.3.
    • If you chose C: That is log (7 ÷ 2). A square root is not a division by 2. It is the power 1/2. Go to M5.2.3.

  5. Which of these is equal to log₂ 9?

    Tap your answer.

    1. Not this one. You turned the fraction upside down. In the change of base formula the number inside the log goes on top: log 9 ÷ log 2. Go to M5.2.3. Try another.

    2. Not this one. That is log (9 ÷ 2) by the quotient law, a different number. Changing the base needs a division of two logs. Go to M5.2.3. Try another.

    3. Yes, that is right. The change of base formula says the log of x to the base a is log x ÷ log a. Here x = 9 and a = 2.

    4. Not this one. Changing the base divides. Check with a number you know: log₂ 8 = 3, and log 8 ÷ log 2 = 0.903 ÷ 0.301 = 3, but log 8 × log 2 is about 0.272. Go to M5.2.3. Try another.

    See the answer

    log 9 ÷ log 2

    The change of base formula says the log of x to the base a is log x ÷ log a. Here x = 9 and a = 2.

    • If you chose A: You turned the fraction upside down. In the change of base formula the number inside the log goes on top: log 9 ÷ log 2. Go to M5.2.3.
    • If you chose B: That is log (9 ÷ 2) by the quotient law, a different number. Changing the base needs a division of two logs. Go to M5.2.3.
    • If you chose D: Changing the base divides. Check with a number you know: log₂ 8 = 3, and log 8 ÷ log 2 = 0.903 ÷ 0.301 = 3, but log 8 × log 2 is about 0.272. Go to M5.2.3.

  6. The lower curve is y = log₂ x. The upper curve is y = log₂ (4x). It is a copy of the lower curve moved straight up. Which equation shows the upper curve as a moved copy?

    A graph with the x axis x and the y axis y. A solid curve named y = log₂ x passes through the point (1, 0) and rises slowly. A dashed curve named y = log₂ (4x) lies above it, passes through the point (1, 2), and has the same shape. A key under the graph names the solid curve y = log₂ x and the dashed curve y = log₂ (4x).12345678−2−112345xyy = log₂ xy = log₂ (4x)
    Tap your answer.

    1. Not this one. That would move the curve up by 4. The graph shows the point (1, 2), so the move is 2, not 4. The 4 is inside the log and becomes log₂ 4. Go to M5.2.3. Try another.

    2. Yes, that is right. The product law gives log₂ (4x) = log₂ 4 + log₂ x. Since 2² = 4, log₂ 4 = 2. So the curve is y = log₂ x + 2, the lower curve moved up by 2.

    3. Not this one. That would stretch the curve upwards, and at x = 1 it would stay at 0. The picture shows the curve at 2 when x = 1. Go to M5.2.3. Try another.

    4. Not this one. That would slide the curve left by 4, and it would not pass through (1, 2). The 4 multiplies x. It does not add to it. Go to M5.2.3. Try another.

    See the answer

    y = log₂ x + 2

    The product law gives log₂ (4x) = log₂ 4 + log₂ x. Since 2² = 4, log₂ 4 = 2. So the curve is y = log₂ x + 2, the lower curve moved up by 2.

    • If you chose A: That would move the curve up by 4. The graph shows the point (1, 2), so the move is 2, not 4. The 4 is inside the log and becomes log₂ 4. Go to M5.2.3.
    • If you chose C: That would stretch the curve upwards, and at x = 1 it would stay at 0. The picture shows the curve at 2 when x = 1. Go to M5.2.3.
    • If you chose D: That would slide the curve left by 4, and it would not pass through (1, 2). The 4 multiplies x. It does not add to it. Go to M5.2.3.

  7. From an earlier chapter

    What is log₂ 0.25?

    Tap your answer.

    1. Not this one. You may have ignored the fraction. 2² = 4, not 0.25. A number below 1 has a negative log. Go to M5.2.2. Try another.

    2. Yes, that is right. 0.25 = 1/4 = 2⁻². So the power of 2 that gives 0.25 is −2.

    3. Not this one. You may have worked from 0.25 = 1/4 and used the 4. The log of 1/4 is the power of 2 that gives 1/4: 2⁻² = 1/4, so the answer is −2. Go to M5.2.2. Try another.

    4. Not this one. You may have taken the square root of 0.25, which is 0.5. A log is the power of 2 that gives 0.25, and 2⁻² = 0.25. Go to M5.2.2. Try another.

    See the answer

    −2

    0.25 = 1/4 = 2⁻². So the power of 2 that gives 0.25 is −2.

    • If you chose A: You may have ignored the fraction. 2² = 4, not 0.25. A number below 1 has a negative log. Go to M5.2.2.
    • If you chose C: You may have worked from 0.25 = 1/4 and used the 4. The log of 1/4 is the power of 2 that gives 1/4: 2⁻² = 1/4, so the answer is −2. Go to M5.2.2.
    • If you chose D: You may have taken the square root of 0.25, which is 0.5. A log is the power of 2 that gives 0.25, and 2⁻² = 0.25. Go to M5.2.2.

  8. Work out log 8 + log 125.

    See the answer

    3

    The product law gives log 8 + log 125 = log (8 × 125) = log 1 000. Since 10³ = 1 000, the answer is 3.

  9. Given that log 2 = 0.30103 and log 3 = 0.47712, find log 54 to 4 decimal places.

    See the answer

    1.7324

    54 = 2 × 3³, so log 54 = log 2 + 3 log 3 = 0.30103 + 3 × 0.47712 = 0.30103 + 1.43136 = 1.73239, which is 1.7324 to 4 decimal places.

  10. Find log₆ 50 to 3 significant figures.

    See the answer

    2.18

    Change the base: log₆ 50 = log 50 ÷ log 6 = 1.69897 ÷ 0.77815 = 2.1834. To 3 significant figures this is 2.18. Check: 6² = 36 and 6³ = 216, and 50 is between them.

  11. Using log 2 = 0.301 and log 3 = 0.477, each to 3 decimal places, find log 1.5.

    Tap your answer.

    1. Yes, that is right. 1.5 = 3 ÷ 2, so the quotient law gives log 1.5 = log 3 − log 2 = 0.477 − 0.301 = 0.176.

    2. Not this one. You took log 2 − log 3. Since 1.5 = 3 ÷ 2, the quotient law gives log 3 − log 2. A number above 1 has a positive log. Go to M5.2.3. Try another.

    3. Not this one. You divided 0.477 by 0.301. The quotient law gives a difference of logs, not a quotient of logs. Go to M5.2.3. Try another.

    4. Not this one. You multiplied 0.301 by 1.5. The number 1.5 is inside the log, so it cannot multiply the log of 2. Go to M5.2.3. Try another.

    See the answer

    0.176

    1.5 = 3 ÷ 2, so the quotient law gives log 1.5 = log 3 − log 2 = 0.477 − 0.301 = 0.176.

    • If you chose B: You took log 2 − log 3. Since 1.5 = 3 ÷ 2, the quotient law gives log 3 − log 2. A number above 1 has a positive log. Go to M5.2.3.
    • If you chose C: You divided 0.477 by 0.301. The quotient law gives a difference of logs, not a quotient of logs. Go to M5.2.3.
    • If you chose D: You multiplied 0.301 by 1.5. The number 1.5 is inside the log, so it cannot multiply the log of 2. Go to M5.2.3.

  12. Given that log 2 = a and log 3 = b, write log 18 in terms of a and b. Give your answer in the form a + ...

    See the answer

    a + 2b

    18 = 2 × 3², so log 18 = log 2 + log (3²) = log 2 + 2 log 3 = a + 2b.

Keep it

Copy this into your exercise book.

log xy = log x + log y. log (x/y) = log x − log y. log (xⁿ) = n log x. Same base. No law for log (x + y). Change of base: the log of x to base a is log x ÷ log a.

Draw: Two bars end to end, log x and log y, and one bar under them for log xy.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (909 KB). It opens on any phone.