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

Combining logs with the three laws

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

  1. Work out log base 10 of 100 plus log base 10 of 10.

    Answer: ______________

  2. Write log x + 2 log y as a single logarithm.

    • log(x y^2)
    • log(x + y^2)
    • 2 log(x y)
    • log(x^2 y)

  3. Write log x + 2 log y as a single logarithm.

    • log(x y^2)
    • log(x + y^2)
    • log(x^2 y)

  4. Nadia says expanding log(a^2 b/c) puts a minus in front of the log c term. Is Nadia right?

    Circle one:   True   False

  5. Use a change of base to work out log base 2 of 8 with common logs. What is its value?

    Answer: ______________

  6. Write 2 log x - log y as a single logarithm.

    • log(x^2 - y)
    • 2 log(x/y)
    • log(x^2/y)

  7. Expand log(a^2 b/c) into separate terms.

    • 2 log a + log b + log c
    • 2 log a + log b - log c
    • log a + 2 log b - log c

  8. Expand log(a^2 b / c) into separate terms.

    • 2 log a + log b - log c
    • 2 log a + log b + log c
    • log a + 2 log b - log c
    • log a + log b divided by log c

  9. Which single logarithm equals 2 log a + log b - log c?

    • log(a^2 + b - c)
    • log(a^2 b/c)
    • log(2ab/c)

  10. Use a change of base to evaluate log base 4 of 8. Give your answer as a decimal.

    Answer: ______________

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Answer key

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

Combining logs with the three laws W1-mt_5vHdWlxaqY-s1

  1. 3 · log base 10 of 100 is 2 and of 10 is 1, and 2 + 1 = 3.
  2. log(x y^2) · 2 log y = log(y^2), and adding logs multiplies: log x + log(y^2) = log(x y^2).
  3. log(x y^2) · Pull the 2 inside to get log(y^2), then the plus fuses the pair by multiplying. That gives log(x y^2).
  4. True · Division inside becomes subtraction outside. The c sits on the bottom, so its log is subtracted.
  5. 3 · Change of base gives log(8)/log(2), and 2^3 = 8, so the value is 3.
  6. log(x^2/y) · Pull the 2 inside to get log(x^2), then the minus divides inside. That gives log(x^2/y).
  7. 2 log a + log b - log c · The square gives 2 log a, the product splits with plus, and the division gives minus log c.
  8. 2 log a + log b - log c · The square gives 2 log a, the product splits with plus, and the division by c gives minus log c.
  9. log(a^2 b/c) · Run expansion backwards: the 2 becomes a power, and plus and minus fuse by multiplying and dividing. That gives log(a^2 b/c).
  10. 1.5 · log base 4 of 8 = log(8)/log(4) = 3/2 = 1.5, since 4^1.5 = 8.
Worksheet · LightMySky