Properties of Logarithms

Section 6.5 Beginning on page 327

PropertiesBecause logarithms are the inverse functions of the exponential functions, properties of logarithms are similar to properties of exponents.

Product Property:

Quotient Property:

Power Property:

log𝑏𝑚𝑛=log𝑏𝑚+ log𝑏𝑛

log𝑏(𝑚𝑛 )=log𝑏𝑚− log𝑏𝑛log𝑏𝑚

𝑛=𝑛 log𝑏𝑚

Using Properties of LogarithmsExample 1: Use and to evaluate each logarithm.

a)

b)

c)

**Use the properties of logarithms to re write the expressions so that you can use the given values to evaluate them.

¿ log23− log 27 ≈1.585−2.807 ≈−1.222

¿ log23+log 27 ≈1.585+2.807≈ 4.392¿ log23 ∙7

¿ log272 ¿2 ∙ log27 ≈2(2.807) ≈5.614

Rewriting Logarithmic ExpressionsYou can use the properties of logarithms to expand and condense logarithmic expressions.

Example 2: Expand

Example 3: Condense

¿ ln5 𝑥7− ln 𝑦 ¿ ln5+ln 𝑥7− ln 𝑦 ¿ ln5+7 ln 𝑥− ln 𝑦

¿ log 9+log 23− log 3 ¿ log 9∙23− log 3 ¿ log 9 ∙23

3 ¿ log 24

Change of Base FormulaLogarithms with bases other than 10 can be written in terms of common or natural logarithms so that you can evaluate any logarithm using a calculator.

and

Example 4: Evaluate using common logarithms.

Example 5: Evaluate using natural logarithms.

log 38=log 8log 3

≈0.90310.4771 ≈1.893

log 624=ln 24ln 6

≈3.17811.7918 ≈1.774

Solving a Real-Life ProblemExample 6: For a sound with intensity I (in watts per square meter), the loudness of the sound (in decibels) is given by the function

where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a tract so that the intensity of the sound doubles. By how many decibels does the loudness increase?

𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑖𝑛𝑙𝑜𝑢𝑑𝑛𝑒𝑠𝑠=𝐿 (2 𝐼 )−𝐿(𝐼 ) ¿10 log2 𝐼𝐼 0−10 log

𝐼𝐼0

¿10 (log 2 𝐼𝐼 0 − log𝐼𝐼0

¿¿)¿10 (log 2+ log 𝐼

𝐼 0− log

𝐼𝐼 0

¿¿)¿10 log2

The loudness increased by decibels, or about 3 decibels.

Monitoring ProgressUse and to evaluate the logarithm.

1) 2) 3) 4)

Expand the logarithmic expression.

5) 6)

Condense the logarithmic expression.

7) 8)

−0.263 2.059 2.322 2.694

¿ log63+4 log6 𝑥 ¿ ln5− ln 12− ln 𝑥

¿ log𝑥9 ¿ ln 9

Monitoring ProgressUse the change-of-vase formula to evaluate the logarithm.

9) 10) 11) 12)

13) For a sound with intensity I (in watts per square meter), the loudness of the sound (in decibels) is given by the function

where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a tract so that the intensity of the sound TRIPLES. By how many decibels does the loudness increase?

≈1.292 ≈1.269 ≈0.674 ≈1.369

10 log 3≈ 4.8𝑑𝑒𝑐𝑖𝑏𝑎𝑙𝑠