unit 3: exponential and logarithmic functions in this activity you will learn how to solve...
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Unit 3: Exponential and Logarithmic Functions
In this activity you will learn how to solve exponential equations using two methods: the common base method
and the logarithmic method.
Activity 6: Solving Exponential Equations
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
Example 1Solve for x:
2x-3 = 8
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
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 8 NO
2
3
4
5
6
Work out the table then click ANSWER to see the answer
x 2x-3 8 Does LS=RS?
1 21-3=2-2=1/4 8 NO
2 ½ 8 NO3 1 8 NO4 2 8 NO5 4 8 NO6 8 8 YES
ANSWER
The solution is x = 6
Since, 26-3=8
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
How can we solve this without trial and error?
1. Write the left side as a power with base 2
2x-3=23
2. Set the exponents as an equation
x – 3 = 3
3. Solve for x
x = 6
4. What do you notice?
You get the same solution as trial and
error
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
Example 2Solve for x:
2x+2 = 4x
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
Using trial and error, let us find the value of x that makes the LS = RS
x 2x+2 4x Does LS=RS?
0
1
2
3
Work out the table then click ANSWER to see the answer
x 2x+2 4x Does LS=RS?
0 4 1 NO1 8 4 NO2 16 16 YES3 32 64 NO
ANSWER
The solution is x = 2
Since, 22+2=24=16=42
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
How can we solve this without trial and error?
1. Write the left side as a power with base 2
2x+2=(22)x =22x
2. Set the exponents as an equation
x + 2 = 2x
3. Solve for x
x = 2
4. What do you notice?
You get the same solution as trial and
error
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
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
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
1. Simplify equation using exponent laws
2(22x)= 122x+1 = 1
2. Write all powers with the same base
22x+1 = 20
3. Simplify algebraically until you have a two power equation LS = RS
22x+1 = 20
4. Set your exponents equal and solve
2x+ 1 = 02x = -1x = -1/2
Example 3Solve: 2(22x)= 1
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
You can check the solution using LS=RS. Below is a graphical check of the solution
Example 3Solve: 2(22x)= 1
x=1/2
y=1
y=2(22x)
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
33x(32)2x-1 = 3x+4
33x(34x-2) = 3x+4
37x-2 = 3x+4
7x – 2 = x + 4x = 1
Example 4Solve: 27x(92x-1) = 3x+4
Unit 3: Exponential and Logarithmic Functions
Method 1: Common Base Method
Activity 6: Solving Exponential Equations
4x(43) + 4x = 1040
64(4x)+ 4x = 1040
65(4x)= 1040
(4x)= 42
.:x = 2
Example 5Solve: 4x+3 + 4x = 1040
(4x)= 16
Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Looking at this equation we can see that 27 cannot be made as a power with a base of 5
52 = 2553 = 125
x must be between 2 and 3 but closer to 2.
Example 6Solve for x:
5x = 27
Let us estimate the value of x
Method 2: Solving with Logarithms
Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
We cannot solve for x since it is an exponent. How can we bring the exponent down so we can solve for it?
Example 6Solve for x:
5x = 27
We can use the logarithmic power law:logbmn = nlogbmon the equation.
Method 2: Solving with Logarithms
Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
log5x = 27
Example 6Solve for x:
5x = 27
Method 2: Solving with Logarithms
Remember, whatever is done on one side must
be done to the other
log5x = log27log5x = log27xlog5 = log27
Use the Laws of Logarithms to bring down the
exponent
Solve the equation by isolating the
variable
log5x = log27xlog5 = log27log5 log 5
Use your calculator to determine log25/log5
log5x = log27xlog5 = log27log5 log 5
x = 2.048
Try another example
Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
Example 7Solve for x:4x = 8(x+3)
Method 2: Solving with Logarithms
log24x = log28(x+3) Set the log to both sides. Use a base of 2 for both
sides.
log24x = log28(x+3)
xlog24 = (x+3)log28log24x = log28(x+3)
xlog24 = (x+3)log28xlog222 = (x+3)log223
Use the Power law of logarithms to bring down the
exponents
Write 4 and 8 as powers of base 2Using the power
property of logarithms the
following occursLogbbm=m
log24x = log28(x+3)
xlog24 = (x+3)log28xlog222 = (x+3)log223
x(2) = (x+3)(3)
Solve the equation by isolating the
variable
log24x = log28(x+3)
xlog24 = (x+3)log28xlog222 = (x+3)log223
2x = 3(x+3)
log24x = log28(x+3)
xlog24 = (x+3)log28xlog222 = (x+3)log223
2x = 3(x+3)2x = 3x+9
log24x = log28(x+3)
xlog24 = (x+3)log28xlog222 = (x+3)log223
2x = 3(x+3)x = -9
Unit 3: Exponential and Logarithmic Functions
Activity 6: Solving Exponential Equations
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