Set up and solve exponential and logarithmic equations

1. 4.5 Exponential & Logarithmic
Equations
Chapter 4 Inverse, Exponential, and Logarithmic
Functions

2. Concepts and Objectives
 Exponential and Logarithmic Equations
 Identify e and ln x.
 Set up and solve exponential and logarithmic
equations.

3. e
 Suppose that $1 is invested at 100% interest per year,
compounded n times per year.
 According to the formula, the compound amount at
the end of 1 year will be
 As n increases, the value of A gets closer to some fixed
number, which is called e.
 e is approximately 2.718281828.
1
1
n
A
n
 
 
 
 

4. Natural Logarithms
 Logarithms with base e are called natural logarithms,
since they often occur in the life sciences and economics
in natural situations that involve growth and decay.
 The base e logarithm of x is written ln x.
 Therefore, e and ln x are inverse functions.
ln
ln
x x
e e x
 

5. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x

6. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x

7. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x
   

 
2 1 2
log3 log0.4
x x

8.   
2 log3 log3 log0.4 2log0.4
x x
Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x
   

 
2 1 2
log3 log0.4
x x

9. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x
   
  
2 1 log3 2 log0.4
x x
  
log0.
l 4
og3
2 log3 2log0.4
x x
  
log0.4
2 log3 2log g
0.4 lo 3
x x

10. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x
   
  
2 1 log3 2 log0.4
x x
  
2 log3 log3 log0.4 2log0.4
x x
 
  
2log3 log0.4 2log0.4 log3
x
  
2 log3 log0.4 2log0.4 log3
x x

11. Exponential Equations
 Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
 

2 1 2
3 0.4
x x
 

2 1 2
log3 log0.4
x x
   
  
2 1 log3 2 log0.4
x x
  
2 log3 log3 log0.4 2log0.4
x x
 
  
2log3 log0.4 2log0.4 log3
x
  
2 log3 log0.4 2log0.4 log3
x x

  

2log0.4 log3
0.236
2log3 log0.4
x

12. Properties of Logs, Revisited
 You could also finish this as



2log0.4 log3
2log3 log0.4
x



2
2
log0.4 log3
log3 log0.4
 

 
 
 
log 0.16 3
9
log
0.4

log0.48
log22.5
This is an
exact answer.

13. Solving a Logarithmic Equation
 Example: Solve    
log 6 log 2 log
x x x
   

14. Solving a Logarithmic Equation
 Example: Solve    
log 6 log 2 log
x x x
   
6
log log
2
x
x
x



6
2
x
x
x



 
6 2
x x x
  
2
6 2
x x x
  
2
6 0
x x
  
  
3 2 0
x x
  
3, 2
x  
  
2
0 6
x x
If we plug in –3, x+2
is negative. We can’t
take the log of a
negative number, so
our answer is 2.

15. Solving a Base e Equation
 Example: Solve and round your answer to the
nearest thousandth.
2
200
x
e 

16. Solving a Base e Equation
 Example: Solve and round your answer to the
nearest thousandth.
2
200
x
e 
2
ln ln200
x
e 
2
ln200
x 
ln200
x  
2.302
x 

17. Solving a Base e Equation
 Example: Solve  
ln
ln ln 3 ln2
x
e x
  

18. Solving a Base e Equation
 Example: Solve  
ln
ln ln 3 ln2
x
e x
  
 
  
ln ln 3 ln2
x
x
 
  
ln
ln ln 3 ln2
x
x
e
ln ln2
3
x
x


2
3
x
x


2 6
x x
 
6 x

 
 
2 3
x x

19. Classwork
 4.5 Assignment (College Algebra)
 Page 464: 12-20 (even), page 442: 28, 30, 60-70
(even), page 429: 80-84 (even)
 4.5 Classwork Check
 Quiz 4.3