1. 4.6 Apply Properties of Logarithms

2. Properties of Logarithms
Let b, u, and v be positive numbers such that b≠1.
Product Property: (addition)
logb (uv) = logb u + logb v
Quotient Property: (subtraction)
logb (u/v) = logb u – logb v
Power Property: (multiplication)
logb (u)n = n logb u

3. Expanding Logarithms Use log53≈.683 and log57≈1.209
Approximate
Approximate

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

5. Expanding Logarithms Completely
Single logarithms with multiple operations can be expanded to multiple logarithms with the same base

6. Expanding Logarithms Completely
Use the properties to expand logarithms.
log27x3 - log2y =
log27 + log2x3 – log2y =
log27 + 3·log2x – log2y

7. 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

8. Condensing Logarithms
Logs with the same base can be combined (condensed) into “one” logarithm

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

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

11. OR Change of Base Formula log37 = log37 log 7 ≈ log 3 ln 7 ≈ ln 3
Use the change of base to evaluate:
When the logarithm is not in base “10”, or base “e”
log37 =
(base 3)
(change to base 10)
log 7 ≈
log 3
(base 10)
1.771
log37
(base 3)
(change to base e)
ln 7 ≈
ln 3
(base e)
1.771 OR

12. Assignment
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