5-6-solving-exponential-and-logarithmic-equations -...

Objectives:
1.
Solve exponential and logarithmic equations.
2.
Solve a variety of application problems by
using exponential and logarithmic equations.
Solving Exponential
and Logarithmic Equations
5.6

Methods of Solving Exponential Equations
1.
Rewrite the bases of the powers on both sides so they are the
same.
.
.

Example #1
Powers of the Same Base
 Solve the equation and confirm your solution with a
graph.
A.
1
3 9


x x
Rewrite both bases to
base 3.
 
1
1 2
3 3
3 3
1 2
1 2
 
 




x
x x
x x
x
x

Example #1
Powers of the Same Base
 Solve the equation and confirm your solution with a
graph.
B.
5
3
4
1
2









x
x
Rewrite to base 2, remember
fractions have negative
exponents.
 
 
 
10 2 3
5
2 3
5
1
2 3
5
1 3
2 2
2 2
2 2
4 2
  

 




 










x x
x
x
x
x
x
x
3
13
13 3
10 3 3
10 2 3
 
 
  
   
x
x
x
x x

Example #1 continued…
Powers of the Same Base
 Solve the equation and confirm your solution with a
graph.
C.
625 25 5
2

x x
Rewrite to base
5.
 
1
4 4
5 5
5 5
5 5 5
5 5 5
4 4
4 2 2
4 2 2
4 2 2







x
x
x
x x
x x
x
x

Example #1 continued…
Powers of the Same Base
 Solve the equation and confirm your solution with a
graph.
D.
x x  
 32 2
6
2
 
  
1 , 6
0 1 6
0 6 5
5 6
2 2
2 2
2
2
5 6
5 6
2
2
  
  
  
  


 


x x
x x
x x
x x
x x
x
x
This time the graphs with the
intersection method were too
difficult to read, so the intercept
method was used to verify the
two solutions.

Example #2
Logarithms on Both Sides
 Solve the equation. Solutions can be confirmed with a
graph.
3 6 
x
Problems like this can be confusing because students will
recognize that 2(3) = 6. Unfortunately, that will not work
because we have to be able to rewrite both sides to the
same base of a power for the first method to be used. Our
only choice then is the second method.
6131 . 0
6 log
3 log
3 log 6 log
3 log 6 log
 
 

x
x
x
6131 . 0
6 ln
3 ln
3 ln 6 ln
3 ln 6 ln
 
 

x
x
x
Notice how common logs and natural logs both work the same
way. Choosing which one to use basically comes down to
personal preference.

 