1. 8.5 Properties of Logarithms

2. Investigation Copy and complete the table.
Use the completed table to write a conjecture about the relationship among logbu, logbv, and logbuv.
logbu
logbv
logbuv
log 10 =
log 100 =
log 1000 =
log 0.1 =
log 0.01 =
log =
log24 =
log28 =
log232 =

3. Properties of Logarithms
Let b, u, and v be positive numbers such that b ≠ 1.
Product Property logb uv = logb u + logb v
Quotient Property
Power Property logb un = n logb u

4. Examples
Use log9 5 ≈ and log9 11 ≈ to approximate the following.
log9 55 =
log9 25 =

5. Examples Expand log5 2x6. Expand Condense 2log3 7- 5log3x
Condense 2log8 x – log8 5 – 3log8 y

6. Change-of-Base Formula
Logarithms with any other base other that10 or e can be written in terms of common or natural logarithms by change-of-base
Change-of-Base Formula
Let u, b, and c be positive numbers with b ≠ 1 and c ≠ 1. Then:
In particular, and

7. Example Evaluate log4 8 using common logarithms.
Evaluate log6 15 using natural logarithms.

8. Example
The Richter magnitude M of an earthquake is based on the intensity I of the earthquake and the intensity I0 of an earthquake that can be barely felt. One formula used is If the intensity of the Los Angeles earthquake in 1994 was times I0, what was the magnitude of the earthquake? What magnitude on the Richter scale does an earthquake have if it intensity is 100 times the intensity of a barley felt earthquake?

9. Example
Use the decibel formula from Example 5 on page How much louder is the sound of four subway trains passing by a point than just one subway trains passing the point?