Mathematical Investigations III
Name:
Mathematical Investigations III
Logarithms
The Third Law of Logarithms
Use a method similar to the ones that were used to prove the First and Second Laws of
Logarithms to prove the Third Law of Logarithms:
THE THIRD LAW OF LOGARITHMS:
 x
log b   = log b  x   log b  y  , b > 0, b  1
 y
Let
= p,
and
=q
Practice:
1.
Rewrite the following using the Third Law of Logarithms and simplify if possible.
a.
b
log b   
5
b.
Logs 6.1
7
log 2   
8
Rev. S11
Mathematical Investigations III
Name:
2.
k 
Prove: log 7    log 7  k   log 7 (11) . Use the method of proof used on page 6.1.
 11 
Putting It All Together
Sometimes two or more laws of logarithms must be used in order to simplify an expression.
Simplify each of the following using whatever laws you need.
3.
 34 
log 2  7  
4 
4.
 3b12 
log b 

 4 
We will learn from subsequent sheets how to find a numerical value for an expression like #3
above. Because we don’t know the base of the logarithm in #4 we can’t find a value for such a
problem. Often problems like this one are asked in a slightly different way (see page 6.3).
Logs 6.2
Rev. S11
Mathematical Investigations III
Name:
Suppose that logb  2  p and logb  3  q . Now you can use the laws of logarithms to show
that
 8b 2 
2
log b 
  log b 8b   log b 81
 81 
 log b  8   log b  b 2   log b  81
 log b  23   log b  b 2   log b  34 
 3log b  2   2 log b  b   4 log b  3
 3 p  2  4q
Additional practice problems:
Suppose that logb  2  p , logb  3  q , or logb  5  r . Give an expression in terms of p, q,
or r for each of the following.
3.
logb  6  
4.
log b 1024b 2  
5.
 25 
log b   
 9 
6.
logb
7.
log b 120b 7  
8.
log 2  b  
9.
Explain why logb  2  3  p  q .
Logs 6.3

3

400 
(CAREFUL!)
Rev. S11