Properties of Logarithms 
 Section 4.5  
Definition of Logs  log _{b }x = y 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  

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} M  
Take
care of multiplication next multiplication turns into addition 
log _{b} MN = log _{b} M + log _{b} N  
Do log _{b}_{ }x^{2 }= 2 log _{b}_{ }x next  log _{b} M ^{p}^{ } = p log _{b} M  
Distribute 1/3 through ( )  

All variables are isolated  

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