1. 8.5 Properties of logarithms

2. 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

3. Use log53≈.683 and log57≈1.209 log53/7 = log521 = log53 – log57 ≈
Approximate:
log53/7 =
log53 – log57 ≈
.683 – =
-.526
log521 =
log5(3·7)=
log53 + log57≈ =
1.892

4. Use log53≈.683 and log57≈1.209
Approximate:
log549 =
log572 =
2 log57 ≈
2(1.209)=
2.418

5. Expanding Logarithms log2 = log27x3 - log2y = log27 + log2x3 – log2y =
You can use the properties to expand logarithms.
log =
log27x3 - log2y =
log27 + log2x3 – log2y =
log27 + 3·log2x – log2y

6. log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x Your turn!
Expand:
log 5mn =
log 5 + log m + log n
log58x3 =
log58 + 3·log5x

7. Condensing Logarithms
log log2 – log 3 =
log 6 + log 22 – log 3 =
log (6·22) – log 3 =
log =
log 8

8. Your turn again! log57 + 3·log5t = log57t3 3log2x – (log24 + log2y)=
Condense:
log57 + 3·log5t =
log57t3
3log2x – (log24 + log2y)=
log2

9. 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)

10. Examples: log37 = log 7 ≈ log 3 1.771 ln 7 ≈ ln 3 1.771 (base 10)
Use the change of base to evaluate:
log37 =
(base 10)
log 7 ≈
log 3
1.771
(base e)
ln 7 ≈
ln 3
1.771

11. Assignment