Solving Exponential Equations with Logarithms · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Freeing x with logarithms

Mathematics · Algebra · ages 17-18
Name ______________________   Date ____________

  1. In solving 3^x = 20, which step frees x from the exponent?

    • Rounding to three significant figures
    • Taking the log of both sides
    • Dividing both sides by 3

  2. A sample halves every 4 hours. Starting from 80 g, how much is left after 4 hours?

    • 40 g
    • 20 g
    • 60 g

  3. Solve 5^x = 125.

    Answer: ______________

  4. Solve 5 to the power x = 125.

    Answer: ______________

  5. Sam substitutes y = e to the x, gets y = 2 or y = 3, then writes x = 2 or x = 3. Is Sam right?

    Circle one:   True   False

  6. A sample halves every 4 hours. Starting from 80 g, how much is left after 12 hours?

    • 20 g
    • 10 g
    • 30 g

  7. A tracer halves every 6 hours. How many hours until 200 MBq decays to 25 MBq?

    Answer: ______________

  8. Solve 3 to the power x = 20, giving x to three significant figures.

    • 2.73
    • 2.63
    • 3.00
    • 2.37

  9. Sam swaps y = e^x, gets y = 2 or y = 3, then writes x = 2 or x = 3. What went wrong?

    • He factored badly and the y values are wrong
    • He rounded too early and lost the third figure
    • He skipped the logs: x = ln 2 or x = ln 3

  10. Solve e to the 2x minus 5 e to the x plus 6 = 0.

    • x = ln 2 or x = ln 3
    • x = 2 or x = 3
    • x = ln 5
    • x = ln 6
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Answer key

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

Freeing x with logarithms W1-mt_DaxxpSbDOn-s1

  1. Taking the log of both sides · The log step pulls the exponent down front through the power rule. That is the step that frees x.
  2. 40 g · One half life has passed, so halve once. Half of 80 g is 40 g.
  3. 3 · 125 is 5^3, so matching exponents gives x = 3.
  4. 3 · 125 is 5 cubed, so matching the exponents gives x = 3.
  5. False · Sam is wrong. Undoing y = e to the x needs logs: x = ln 2 or x = ln 3.
  6. 10 g · 12 hours holds 3 half lives, so halve three times. That gives 80 to 40 to 20 to 10 g.
  7. 18 · 200 to 100 to 50 to 25 is 3 halvings, and 3 times 6 hours = 18 hours.
  8. 2.73 · Taking logs gives x = log 20 over log 3, which is about 2.7268, so 2.73 to three significant figures.
  9. He skipped the logs: x = ln 2 or x = ln 3 · Undoing y = e^x needs natural logs. So x = ln 2 or x = ln 3, and Sam is wrong.
  10. x = ln 2 or x = ln 3 · Substituting y = e to the x gives y squared minus 5y plus 6 = 0, so y = 2 or y = 3, meaning x = ln 2 or x = ln 3.
Worksheet · LightMySky