Three Laws of Logarithms

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Many standardized tests will challenge you with these three laws of logarithms.

Law 1: The logarithm of a product is equal to the sum of the logarithms of each term in the product.

log b xy = log b x + log b y

  • log2 3(4) =
    log2 3 + log2 4
  • log7 5(6) =
    log7 5 + log7 6
  • log15 8(12) =
    log15 8 + log15 12
  • log11 9(7) =
    log11 9 + log11 7

Law 2: The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

log b x/y = log b x – log b y

  • log2 5/6 =
    log2 5 - log2 6
  • log3 7/8 =
    log3 7 - log3 8
  • log5 3/4 =
    log5 3 - log5 4
  • log7 6/11 =
    log7 6 - log7 11

Law 3: The logarithm of x with a rational exponent is equal to the exponent times the logarithm.

log b xn = n log b x

  • log2 56 =
    6 log2 5
  • log3 78 =
    8 log3 7
  • log5 123 =
    3 log5 12
  • log9 97 =
    7 log9 9
  • log3 71/4 =
    1/4 log3 7
  • log9 97/3 =
    7/3 log9 9