8.5 Properties of Logarithms

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 0.001 = log24 = log28 = log232 =

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

Examples Use log9 5 ≈ 0.732 and log9 11 ≈ 1.091 to approximate the following. log9 55 = log9 25 =

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

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

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

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 106.8 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?

Example Use the decibel formula from Example 5 on page 495. How much louder is the sound of four subway trains passing by a point than just one subway trains passing the point?