# 8 5 Properties of logarithms by 14UBD9R5

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Properties of logarithms
p. 493
Properties of Logarithms
• Let b, u, and v be positive numbers such
that b≠1.
• Product property:
• logbuv = logbu + logbv
• Quotient property:
• logbu/v = logbu – logbv
• Power property:
• logbun = n logbu
Use log53≈.683 and log57≈1.209
• Approximate:
• log53/7 =         •log521 =
• log53 – log57 ≈   •log5(3·7)=
•log53 + log57≈
• .683 – 1.209 =
•.683 + 1.209 =
• -.526             •1.892
Use log53≈.683 and log57≈1.209
• Approximate:

• log549 =
• log572 =
• 2 log57 ≈
• 2(1.209)=
• 2.418
Expanding Logarithms
• You can use the properties to expand logarithms.

3
7x
• log2    =
y
• log27x3 - log2y =
• log27 + log2x3 – log2y =
• log27 + 3·log2x – log2y
• Expand:
• log 5mn =
• log 5 + log m + log n

• Expand:
• log58x3 =
• log58 + 3·log5x
Condensing Logarithms
• log 6 + 2 log2 – log 3 =
• log 6 + log 22 – log 3 =
• log (6·22) – log 3 =

62 2
• log     =
3

• log 8
• Condense:
• log57 + 3·log5t =
• log57t3
• Condense:
• 3log2x – (log24 + log2y)=
3
x
• log2
4y
Change of base formula:
• u, b, and c are positive numbers with b≠1 and c≠1. Then:

log b u
• logcu =
log b c
log u
• logcu =                 (base 10)
log c

• logcu = ln u            (base e)
ln c
Examples:
• Use the change of base to evaluate:
• log37 =
• (base 10)          •(base e)
• log 7 ≈            •ln 7 ≈
• log 3              •ln 3
•1.771
• 1.771
Assignment

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