Properties of Logarithms

Properties of Logarithms

What is a logarithm?

• A logarithm is another way to express an exponent

• A log function is the inverse of an exponential function

• The log of a number "X" to a certain base "a" is the same as saying what is the exponent that I need to raise the base to, to get the value "x"

• If it is not specified, the base is 10

Product Property

Example

Product Property

Product Property

Example

Quick example of the Product Property of logarithms.

Multiple Choice

Condense $$\log_35+\log_34$$  

• $$\log_39$$  

• $$\log_320$$  

• $$\log_69$$  

• $$\log_620$$  

Multiple Choice

Condense

$$\log_53+\log_5y+\log_5z$$  

• $$\log_53yz$$  

• $$\log_{15}3yz$$  

• $$\log_35yz$$  

• $$\log_z15y$$  

Multiple Choice

Expand

$$\log_28y$$  

• $$\log_82+\log_8y$$  

• $$\log_28+\log_2y$$  

• $$\log_2y+\log_82$$  

• $$\log_y2+\log_y8$$  

Multiple Choice

Solve $$\log_73+\log_7y=\log_718$$  

• 6

• 54

• 1/6

• 21

Multiple Choice

Solve $$\log_{10}\left(3m-5\right)+\log_{10}m=\log_{10}2$$  

• -5

• 7/4

• 2, -1/3

• -2, 1/3

Quotient Property

Quotient Property

Example

Quick example of the Quotient Property of Logarithms

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

Condense $$\ln5-\ln r$$  

• $$\ln\frac{e}{5}$$  

• $$\ln\frac{e}{r}$$  

• $$\frac{\ln r}{5}$$  

• $$\ln\frac{5}{r}$$  

Multiple Choice

Expand $$\log_4\frac{y}{10}$$  

• $$\log_4y+\log_410$$  

• $$\log_410-\log_4y$$  

• $$\log_{10}y-\log_{10}4$$  

• $$\log_4y-\log_410$$  

Multiple Choice

Solve

$$\log_848-\log_8y=\log_84$$  

• 1/12

• 12

• 44

• 192

Multiple Choice

Solve

$$\log_8\left(t+10\right)-\log_8\left(t-1\right)=\log_812$$  

• 1

• 2

• 11

• 22

Power Property

Power Property

Example

Quick example of the Power Property of Logarithms

Multiple Choice

Condense

$$3\log_47$$  

• $$\log_47^3$$  

• $$\log_421$$  

• $$\log_43^7$$  

• $$\log_410$$  

Multiple Choice

Expand

$$\log_2^{ }y^5$$  

• $$y\cdot\log_25$$  

• $$5\log_2y$$  

• $$2\cdot\log_5y$$  

• $$\log_25^y$$  

Multiple Choice

Solve

$$3\log_74=2\log_7b$$  

• 64

• 4

• 8

• 16

Multiple Choice

Solve

$$3\log_82-\log_84=\log_8g$$  

• 2

• 4

• 8

• 12

Multiple Choice

Solve

$$\log_2w+2\log_25=0$$  

• 0

• 10

• 25

• 32

Multiple Choice

Solve

$$\log_3y=-\log_316+\frac{1}{3}\log_364$$  

• 4

• 16

• 1/4

• 1/2

Multiple Choice

Solve

$$\ln5x+\ln3x=9$$  

• 3/5

• 540

• 22.2324

• 23.2423