Properties of Logarithms

-- Section 4.5 --

 
Definition of Logs

log b x =  Û x = b y

A logarithm is an exponent.

Write in terms of x, y, and z. *   
We want to write the expression such that the variables are as isolated as possible.

Solve

            Step

    Property

Start with division 
division turns into subtraction

     =  log 3 + log x - (log y + log z)

Take care of multiplication next 
multiplication turns into addition
log b MN = log b M + log b N
     =  log 3 + log x - log y - log z Distribute the minus sign  Recall: log  x = log 10 x  

 

All variables are isolated


* Assume all values for x, y, and z are defined for each expression. Example log (-2) is undefined.

  
Write in terms of the logarithms  of x, y, and z

[Solution]

Write in terms of the logarithms  of x, y, and z
 
We want to write the expression such that the variables are as isolated as possible.

Solve

            Step

    Property

Start with division 
division turns into subtraction
Recall:  Convert radicals into exponents

Replace radicals
Exponents turn into multiplication log b M p  = p log b
Take care of multiplication next 
multiplication turns into addition
log b MN = log b M + log b N
Do  log b x2 = 2 log b x next log b M p  = p log b
Distribute 1/3 through ( ) 

 

All variables are isolated


* Assume all values for x, y, and z are defined for each expression. Example log (-2) is undefined.

 

Write in terms of the logarithms  of x, y, and z

[Solution]

Evaluate the following to 3 decimals places:    log 2 5

Solve

            Step

    Property

Change of Base formula
HINT: use these key strokes  log 5 log 2 [enter or =]

Evaluate the following to 3 decimals places:    log 5 2

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada