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Logarithms and their laws Cambridge IGCSE Additional Mathematics revision

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In plain words

A logarithm answers one question: what power? log₂ 8 asks "2 to what power gives 8?", and the answer is 3. Everything else about logs follows from the rules for powers.

7 things to know

  1. log_a x = y means a^y = x. The log is the power.
  2. lg means log to base 10. ln means log to base e.
  3. Adding logs multiplies the numbers: log x + log y = log(xy).
  4. Subtracting logs divides: log x − log y = log(x/y).
  5. A number in front becomes a power: n log x = log(xⁿ).
  6. For any base a: log_a a = 1 and log_a 1 = 0.
  7. Change of base: log_a b = (log_c b) ÷ (log_c a). A useful special case: log_a b = 1 ÷ log_b a.

Worked example

Write 3 + 2 lg p − lg q as a single logarithm.

  1. Turn the 3 into a log to base 10: 3 = lg 1000.
  2. 2 lg p = lg p².
  3. lg 1000 + lg p² − lg q = lg(1000p²/q).

Worked example

Solve log₃(x + 6) − log₃ x = 2.

  1. Combine the logs: log₃((x + 6)/x) = 2.
  2. Undo the log: (x + 6)/x = 3² = 9.
  3. x + 6 = 9x, so x = 3/4.

Tips and tricks

  • The laws only work for logs with the same base. And log(x + y) is not log x + log y.
  • To get rid of a log, raise the base to the power of the other side: log₃ A = 2 means A = 3².
6 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Logarithms and their laws: 6 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is log₂ 8?
    • 2
    • 3 (the answer)
    • 4
    • 16

    2³ = 8.

  2. What is log a + log b as a single logarithm?
    • log(a + b)
    • log(ab) (the answer)
    • log(a/b)
    • (log a)(log b)

    Adding logs multiplies the numbers.

  3. What is 3 log x as a single logarithm?
    • log 3x
    • log x³ (the answer)
    • log(x + 3)
    • (log x)³

    The number in front becomes a power.

  4. What is log₅ 1?
    • 0 (the answer)
    • 1
    • 5
    • undefined

    Any base to the power 0 is 1.

  5. What is ln e?
    • 0
    • 1 (the answer)
    • e
    • 2.718

    e to the power 1 is e.

  6. What is log₉ 3?
    • 2
    • 3
    • 1/2 (the answer)
    • 1/3

    9 to the power 1/2 is 3.

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