Math 141
4.4 – Laws of Logarithms
Warnock - Class Notes
Proofs:
Law 1: Let loga A  u and loga B  v , then ____________ and ________________
 A  
Law 2: Let log a A  log a   B 
 B  


and use law 1.
Law 3: Let loga A  u . Then ________________
 
then, loga AC 
1. Evaluate
a) log50  log200 
b) log2 60  log215 
c)
1 log 81 
4 3
(two ways)
d) log5 5 
2. Use Laws of Logarithms to expand.
a)
log2  xy  
10


b) log2 AB2 
 3 4



c)
ln  x 6y
e

d)
log 4 x2  y2 
3. Combine the following logarithms.
a)
2log x  3log  x 1 




b) ln a  b  ln a  b  2ln c 
log  x  y 
CAUTION!!!
log6
log2
 log x  log y
 log6  log2

log2 x
log6
log2

log 6
2
  3log2 x
3
4. Wealth Distribution Vilfredo Pareto (1848–1923) observed that most of the wealth of a
country is owned by a few members of the population. Pareto’s Principle is
log P  log c  k logW
where W is the wealth level (how much money a person has) and P is the number of people in the
population having that much money.
(a)Solve the equation for P.
(b) Assume that k=2.1 and c=8000, and that W is measured in millions of dollars. Use part (a) to find the
number of people who have $2 million or more. How many people have $10 million or more?
Lets look at
y  logb x
Exponential form
Take
loga of both sides 
Log properties 
Solve for
y
by  x
 
loga b y  loga x
y loga b  loga x
y  loga x
loga b
(remember,
y  logb x )
This gives us our
5. Use the Change of Bass formula and common or natural logs to evaluate
a) log4 10 
b) log7 50 
6. Graph f  x   log5 x on the graphing calculator.
4.4 – Laws of Logarithms
Math 141
Practice
Warnock - Class Notes
1. Evaluate using laws of exponents
a) log2 40  log2 15  log2 6 
c) log3 27 
2. Use Laws of Logarithms to expand completely.
 3
e
ln 



2
 y  z   

x5


3. Use the Change of Bass formula and common or natural logs to evaluate
log8 30 