9.5 Intro: Properties of Logarithms

Properties of Logarithms

After studying this section you should be able to use the product rule, use the quotient rule, use the power rule and expand logarithmic expressions.

Quick example of the Product Property of Logarithms.

Product Property

Product Property: Expand Example

Multiple Select

Expand: $$\log_5\left(xy\right)$$  

• $$\log_5x\ -\ \log_5y$$  

• $$x\log_5y$$  

• $$\log_5x\ +\ \log_5y$$  

• $$y\log_5x$$  

Multiple Select

$$Expand:\ \ \ \log_2\left(a\cdot b\right)$$  

• $$\log_2b-\log_2a$$  

• $$\frac{\log_2a}{2}$$  

• $$1+\log_2a$$  

• $$\log_2a+\log_2b$$  

Multiple Choice

• A

• B

• C

• D

Multiple Choice

$$Expand:\ \ \ \ \log_8\left(3\cdot2\cdot11\right)$$  

• $$2\log_83$$  

• $$6\log_83+6\log_82$$  

• $$\log_83+\log_82+\log_811$$  

Multiple Select

$$Expand:\ \log_9\left(8\cdot5\cdot11\right)$$  

• $$\log_98+4\log_95$$  

• $$\log_98+\log_95+\log_911$$  

• $$2\log_98$$  

• $$4\log_98-\log_95$$  

Product Property: Condense Example

Multiple Choice

Condense: $$\log_35+\log_34$$  

• $$\log_39$$  

• $$\log_320$$  

• $$\log_69$$  

• $$\log_620$$  

Multiple Choice

Condense:

$$\log_53+\log_5y+\log_5z$$  

• $$\log_53yz$$  

• $$\log_{15}3yz$$  

• $$\log_35yz$$  

• $$\log_z15y$$  

Quick example of the Quotient Property of Logarithms

Quotient Property

Quotient Property: Condense Example

Multiple Choice

Expand: $$\ln\left(\frac{u}{v}\right)$$  

• $$\ln\ u\ +\ \ln\ v$$  

• $$v\ln u$$  

• $$4\ln u$$  

• $$\ln u-\ln v$$  

Multiple Choice

Expand

• A

• B

• C

• D

Multiple Choice

Expand:  $$\log_7u^3$$  

• $$\log_7u-\log_7v$$  

• $$u\log_7v$$  

• $$3\log_7u$$  

Multiple Choice

Condense: $$\ln5-\ln r$$  

• $$\ln\frac{e}{5}$$  

• $$\ln\frac{e}{r}$$  

• $$\frac{\ln r}{5}$$  

• $$\ln\frac{5}{r}$$  

Multiple Choice

Condense $$\log_35+\log_34$$  

• $$\log_39$$  

• $$\log_320$$  

• $$\log_69$$  

• $$\log_620$$  

Multiple Choice

Condense

$$\log_7y-\log_79$$  

• $$\log_7\frac{9}{y}$$  

• $$\log_79y$$  

• $$\log_7\frac{y}{9}$$  

• $$\log_9\frac{y}{7}$$  

Multiple Choice

Expand: $$\log_4\frac{y}{10}$$  

• $$\log_4y+\log_410$$  

• $$\log_410-\log_4y$$  

• $$\log_{10}y-\log_{10}4$$  

• $$\log_4y-\log_410$$  

Quick example of the Power Property of Logarithms

Power Property

​Log₅x⁶ = 6Log₅x

Multiple Choice

Expand: $$\log_2^{ }y^5$$  

• $$y\cdot\log_25$$  

• $$5\log_2y$$  

• $$2\cdot\log_5y$$  

• $$\log_25^y$$  

Multiple Choice

Condense: $$3\log_47$$  

• $$\log_47^3$$  

• $$\log_421$$  

• $$\log_43^7$$  

• $$\log_410$$  

Poll

How do you feel about this concept? (Properties of Logarithms)

• BAD

• Okay, need more practice though.

• AWESOME