8.5Properties 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 =

• log53 – log57 ≈

• .683 – 1.209 =

• -.526

•log521 =•log5(3·7)=•log53 + log57≈•.683 + 1.209 =•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.

• log2 =

• log27x3 - log2y =

• log27 + log2x3 – log2y =

• log27 + 3·log2x – log2y

y

x37

Your turn!• 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 =

• log =

• log 8

3

26 2

Your turn again!

• Condense:

• log57 + 3·log5t =• log57t3

• Condense:

• 3log2x – (log24 + log2y)=

• log2 y

x

4

3

Change of base formula:

• u, b, and c are positive numbers with b≠1 and c≠1. Then:

• logcu =

• logcu = (base 10)

• logcu = (base e)

c

u

b

b

log

log

c

u

log

log

c

u

ln

ln

Examples:

• Use the change of base to evaluate:

• log37 =• (base 10)

• log 7 ≈

• log 3

• 1.771

•(base e)•ln 7 ≈ •ln 3•1.771

Assignment