The Laws of Logarithms · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Combining logs with the three laws

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

  1. Use the log laws to evaluate log(1000) - 2 log(10). Here log means base 10.

    Answer: ______________

  2. Which expression can be combined directly into one log using the product law, without changing a base?

    • log base 2 of 5 + log base 3 of 7
    • log base 2 of 5 - log base 2 of 7
    • log base 3 of 5 + log base 3 of 7
    • (log base 2 of 5) times (log base 2 of 7)

  3. For x > 0 and y > 0, write log x + 2 log y as a single logarithm. Here log means base 10.

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

  4. Nadia says log(a+b) = log a + log b for every a > 0 and b > 0. Here log means base 10. Is her statement true?

    Circle one:   True   False

  5. Using change of base, log base 7 of 11 = ln(11)/ln(n). What is n?

    Answer: ______________

  6. For x > 0 and y > 0, write 2 log x - log y as a single logarithm. Here log means base 10.

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

  7. For a > 0, b > 0 and c > 0, expand log(a^2 b/c). Here log means base 10.

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

  8. For positive m, n and p, expand log((mn)^2/p^3). Here log means base 10.

    • 2 log m + 2 log n - 3 log p
    • 2 log m + log n - 3 log p
    • 2 log m + 2 log n + 3 log p
    • log m + 2 log n - 3 log p

  9. For positive x, y and z, which single logarithm equals 3 log x - 2 log y - log z? Here log means base 10.

    • log((x^3 - y^2)/z)
    • log(x^3/(y^2 z))
    • log(x^3 z/y^2)

  10. Given log base 2 of u = 5 and log base 2 of v = 2, use change of base to evaluate log base v of u. 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. 1 · The power law gives 2 log(10) = log(100). The quotient law gives log(1000/100) = log(10) = 1.
  2. log base 3 of 5 + log base 3 of 7 · The product law combines added logs with the same base and positive arguments. The two added base-3 logs meet these conditions.
  3. log(x y^2) · The power law gives 2 log y = log(y^2). The product law then gives log(x y^2).
  4. False · The product law combines the sum as log(ab), not log(a+b). At a = b = 10, the two sides are log(20) and 2, which are different.
  5. 7 · The denominator is the natural log of the original base, 7. Therefore n = 7.
  6. log(x^2/y) · The power law gives log(x^2). The quotient law then divides x^2 by y, giving log(x^2/y).
  7. 2 log a + log b - log c · The quotient law subtracts log c. The product law separates a^2 and b. The power law gives 2 log a.
  8. 2 log m + 2 log n - 3 log p · The square applies to the whole product mn, so both log m and log n receive a factor of 2. The denominator contributes -3 log p.
  9. log(x^3/(y^2 z)) · The coefficients give powers x^3 and y^2. Both subtracted logs supply denominator factors, giving log(x^3/(y^2 z)).
  10. 2.5 · Change to base 2: log base v of u = (log base 2 of u)/(log base 2 of v) = 5/2 = 2.5.
Worksheet · LightMySky