Logarithms as the Inverse of Exponentials · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Logs undo powers

Mathematics · Algebra · ages 16-17
Name ______________________   Date ____________

  1. What is log base 2 of 32?

    Answer: ______________

  2. Which logarithm form matches 2^5 = 32?

    • log base 2 of 32 = 5
    • log base 5 of 32 = 2
    • log base 32 of 2 = 5
    • log base 2 of 5 = 32

  3. What is log base 2 of 32?

    • 16
    • 5
    • 4

  4. Log base 2 of a negative number has a value.

    Circle one:   True   False

  5. Exponentials and logarithms undo each other.

    Circle one:   True   False

  6. What is log base 10 of 1000?

    Answer: ______________

  7. Why does the logarithm of a negative number have no value?

    • Because 2^x is always positive, so its inverse cannot take a negative input
    • Because negative numbers are too large for logs
    • Because logs only work on whole numbers
    • Because zero and negatives break calculators only

  8. Why can ln x never accept zero?

    • e to the x never outputs zero, so the swap leaves zero out
    • Zero is too small to write
    • Logs skip round numbers

  9. A sketch shows ln x crossing (0, 1) like e to the x. What is wrong?

    • Crossings never move
    • The axes were labelled wrongly
    • The reflection swaps the point to (1, 0)

  10. A learner rewrites log base 3 of 81 as 3 to the 81. What slipped?

    • 81 is not a power of 3
    • Logs cannot be rewritten
    • Power and log forms were swapped the wrong way round
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Logs undo powers W1-mt_skLQLUan0T-s1

  1. 5 · 2^5 = 32, so log base 2 of 32 = 5.
  2. log base 2 of 32 = 5 · The base stays 2, the power 5 becomes the answer: log base 2 of 32 = 5.
  3. 5 · 2 to the 5 is 32, so the missing power is 5.
  4. False · No power of 2 lands below zero, so the question has no answer.
  5. True · One direction powers up, the other returns the power.
  6. 3 · 10^3 = 1000, so log base 10 of 1000 = 3.
  7. Because 2^x is always positive, so its inverse cannot take a negative input · The exponential never outputs negatives, so the logarithm, its inverse, cannot accept them as inputs.
  8. e to the x never outputs zero, so the swap leaves zero out · Inverses swap inputs and outputs, and zero is never an output.
  9. The reflection swaps the point to (1, 0) · Mirroring moves every point, including the crossing.
  10. Power and log forms were swapped the wrong way round · The 81 is the result, not the power; the power is 4.
Worksheet · LightMySky