1. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
In this activity you will learn how to solve exponential equations using two methods: the common base method and the logarithmic method.

2. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 1
Solve for x:
2x-3 = 8

3. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Using trial and error, let us find the value of x that makes the LS = RS x
2x-3 8
Does LS=RS? 1
21-3=2-2=1/4 NO 2 3 4 5 6
YES x
2x-3 8
Does LS=RS? 1
21-3=2-2=1/4 NO 2 3 4 5 6
Work out the table then click ANSWER to see the answer
ANSWER
The solution is x = 6
Since, 26-3=8

4. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
How can we solve this without trial and error?
1. Write the right side as a power with base 2
3. Solve for x
2x-3=23
x = 6
2. Since the bases are the same, the exponents must be equal.
4. What do you notice?
You get the same solution as trial and error
x – 3 = 3

5. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 2
Solve for x:
2x+2 = 4x

6. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Using trial and error, let us find the value of x that makes the LS = RS x
2x+2 4x
Does LS=RS? 4 1 NO 8 2 16
YES 3 32 64 x
2x+2 4x
Does LS=RS? 1 2 3
Work out the table then click ANSWER to see the answer
ANSWER
The solution is x = 2
Since,
22+2=24=16=42

7. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
How can we solve this without trial and error?
1. Write the right side as a power with base 2
3. Solve for x
2x+2=(22)x =22x
x = 2
2. Since the bases are the same, the exponents must be equal.
4. What do you notice?
You get the same solution as trial and error
x + 2 = 2x

8. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Process
1. Simplify equation using exponent laws
2. Write all powers with the same base
3. Simplify algebraically until you have a two power equation LS = RS
4. Set your exponents equal and solve

9. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 3
Solve: 2(22x)= 1
1. Simplify equation using exponent laws
3. Simplify algebraically until you have a two power equation LS = RS
2(22x)= 1
22x+1 = 1
22x+1 = 20
4. Set your exponents equal and solve
2. Write all powers with the same base
2x+ 1 = 0
2x = -1
x = -1/2
22x+1 = 20

10. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 3
Solve: 2(22x)= 1
You can check the solution using LS=RS. Below is a graphical check of the solution
y=2(22x)
x=-1/2
y=1

11. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 4
Solve: 27x(92x-1) = 3x+4
33x(32)2x-1 = 3x+4
33x(34x-2) = 3x+4
37x-2 = 3x+4
7x – 2 = x + 4
x = 1

12. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 1: Common Base Method
Example 5
Solve: 4x+3 + 4x = 1040
4x(43) + 4x = 1040
64(4x)+ 4x = 1040
65(4x)= 1040
(4x)= 16
(4x)= 42
.:x = 2

13. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 2: Solving with Logarithms
Example 6
Solve for x:
5x = 27
Looking at this equation we can see that 27 cannot be made as a power with a base of 5
Let us estimate the value of x
52 = 25
53 = 125
x must be between 2 and 3 but closer to 2.

14. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 2: Solving with Logarithms
Example 6
Solve for x:
5x = 27
We cannot solve for x since it is an exponent. How can we bring the exponent down so we can solve for it?
We can use the logarithmic power law:
logbmn = nlogbm
on the equation.

15. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 2: Solving with Logarithms
Example 6
Solve for x:
5x = 27
log5x = log27
xlog5 = log27
log5 log 5
x = 2.048
log5x = log27
xlog5 = log27
log5 log 5
log5x = log27
xlog5 = log27
log5x = log27
log5x = 27
Try another example
Use your calculator to determine log25/log5
Remember, whatever is done on one side must be done to the other
Solve the equation by isolating the variable
Use the Laws of Logarithms to bring down the exponent

16. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Method 2: Solving with Logarithms
Example 7
Solve for x:
4x = 8(x+3)
log24x = log28(x+3)
xlog24 = (x+3)log28
xlog222 = (x+3)log223
2x = 3(x+3)
log24x = log28(x+3)
xlog24 = (x+3)log28
xlog222 = (x+3)log223
2x = 3(x+3)
2x = 3x+9
log24x = log28(x+3)
xlog24 = (x+3)log28
xlog222 = (x+3)log223
2x = 3(x+3)
x = -9
log24x = log28(x+3)
xlog24 = (x+3)log28
xlog222 = (x+3)log223
x(2) = (x+3)(3)
log24x = log28(x+3)
xlog24 = (x+3)log28
xlog222 = (x+3)log223
log24x = log28(x+3)
xlog24 = (x+3)log28
log24x = log28(x+3)
Set the log to both sides. Use a base of 2 for both sides.
Solve the equation by isolating the variable
Using the power property of logarithms the following occurs
Logbbm=m
Use the Power law of logarithms to bring down the exponents
Write 4 and 8 as powers of base 2

17. Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
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