Properties of logarithms

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Properties of logarithms. Math 3 Keeper 29. Properties of Logarithms. Let b, u, and v be positive numbers such that b ≠1. Product property: log b uv = log b u + log b v Quotient property: log b u/v = log b u – log b v Power property: log b u n = n log b u. - PowerPoint PPT Presentation

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  • Properties of logarithmsMath 3 Keeper 29

  • Properties of LogarithmsLet b, u, and v be positive numbers such that b1.Product property:logbuv = logbu + logbvQuotient property:logbu/v = logbu logbvPower property:logbun = n logbu

  • EXAMPLE 1: Use log53.683 and log571.209 to approximate the following: a) log53/7 =log53 log57 0.683 1.209 = -0.526

    b) log521 = log5(37)= log53 + log570.683 + 1.209 = 1.892

  • EXAMPLE 1 (continued): Use log53.683 and log571.209 to approximate the following:c) log549 = log572 = 2 log57 2(1.209)= 2.418

  • YOUR TURN!Use log950.732 and log9111.091 to approximate the following: d) log95/11 e) log955

    f) log925

    d) -0.359 e) 1.823

    f) 1.464

  • EXAMPLE 2: Expand the given logarithm*You can use the properties to expand logarithms.

    a) log2 =

    log27x3 - log2y = log27 + log2x3 log2y = log27 + 3log2x log2y

  • Your turn!Expand the logarithm.b) log 5mn = log 5 + log m + log n

    c) log58x3 = log58 + 3log5x

  • EXAMPLE 3: Condense the logarithmsa) log 6 + 2 log2 log 3 = log 6 + log 22 log 3 = log (622) log 3 = log =

    log 8

  • YOUR TURN AGAIN! Condense the logarithm.b) log57 + 3log5t = log57t3

    c) 3log2x (log24 + log2y)=

    log2

  • Change of base formula:u, b, and c are positive numbers with b1 and c1. Then:

    logcu =

    logcu = (base 10)

    logcu = (base e)

  • EXAMPLE 4: Evaluate using the change-of-base formula using 1) common log & 2) natural logarithm.a1) log37 = log 7 log 3 0.8451 0.4771 1.771a2) ln 7 ln 3 1.946 1.099 1.771

  • EXAMPLE 4 (continued): Evaluate using the change-of-base formula using 1) common log & 2) natural logarithm.b1) log48 = log 8 log 4 0.903 0.602 1.500a2) ln 8 ln 4 2.079 1.386 1.500