The document discusses exponential and logarithmic equations. It defines exponential growth using the formula A = P(1 + r/n)^nt where A is the final amount, P is the principal amount, r is the annual interest rate, t is time in years, and n is the number of times interest is compounded per year. As n increases without limit, the value of A approaches a fixed number called e, which is approximately 2.71828. Natural logarithms use e as the base and are written as ln(x). The document provides examples of solving exponential and logarithmic equations by changing between exponential and logarithmic form, isolating the variable, and applying logarithmic properties. It concludes with assigning practice problems from

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