1. and Logarithmic Equations
Section 12.6
Exponential
and Logarithmic Equations
Phong Chau

2. Properties
Power Rule:

3. Exponential Equations
Equations with variables in the exponents are called Exponential Equations
For simple equation, use the followingprinciple:

4. Solving simple equations
These exponential equations are simple because we can express both sides of the equations as a power of the same base

5. Solving exponential equation
Take the natural logarithm on both sides
Use Power Rule
Divide both sides by ln 2
This is the exact solution
Use calculator to find the approximate solution

6. Strategies for Solving Exponential Equations
Isolate the exponential term
Take the natural logarithm on both sides
Use the power rule to pull the x out of the exponent
Solve the resulting equation
Check the answer in the original equation.

7. Example Solve: e1.32t – 2000 =0 Solution We have: e1.32t = 2000
Take the natural logarithm
ln e1.32t = ln 2000
1.32t = ln 2000
t = (ln 2000)/1.32

8. Example Solve: 3 x +1 – 43 = 0 Solution We have 3 x +1 = 43
log 3 x +1 = log 43
(x +1)log 3 = log 43
Power rule for logs
x +1 = log 43/log 3
x = (log 43/log 3) – 1
The solution is (log 43/log 3) – 1, or approximately

9. Example

10.
The above link is a good website to learn new concept. It has many applications.

11. World Population
The world population in billions at time t, where t = 0 represents the year 2000, is given by:
When will the population reach 12 billions?

12. Strategies for solving Logarithmic Equations
Move all logarithms to left hand side (LHS)
Write the LHS as a single logarithm
Rewrite the equation in exponential form
Solve the resulting equation
Check the answer in the original equation.

13. Solution Example Solve: log2(6x + 5) = 4. log2(6x + 5) = 4 6x + 5 = 24
Check the solution!

14. Example Solve: log x + log (x + 9) = 1. Solution
log[x(x + 9)] = 1
x(x + 9) = 101
x2 + 9x = 10
x2 + 9x – 10 = 0
(x – 1)(x + 10) = 0
x – 1 = 0 or x + 10 = 0
x = 1 or x = –10

15. Check x = 1: log 1 + log (1 + 9) 0 + log (10) 0 + 1 = 1 x = –10:
= 1
TRUE
x = –10:
log (–10) + log (–10 + 9)
FALSE
The logarithm of a negative number is undefined. The solution is 1.

16. Example Solution Solve: log3(2x + 3) – log3(x – 1) = 2.
log3[(2x + 3)/(x – 1)] = 2
(2x + 3)/(x – 1) = 32
(2x + 3)/(x – 1) = 9
(2x + 3) = 9(x – 1)
2x + 3 = 9x – 9
x = 12/7
Check the solution!

17. Examples

18. Examples

19. Group Exercise