Unit 3: Exponential and Logarithmic Functions

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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.

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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations

• Method 1 Common Base Method

Example 1 Solve for x 2x-3 8
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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 LSRS?
1 21-32-21/4 8 NO
2
3
4
5
6
x 2x-3 8 Does LSRS?
1 21-32-21/4 8 NO
2 ½ 8 NO
3 1 8 NO
4 2 8 NO
5 4 8 NO
6 8 8 YES
Work out the table then click ANSWER to see the
answer
ANSWER
The solution is x 6 Since, 26-38
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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 left side as a power with base 2
3. Solve for x
2x-323
x 6
2. Set the exponents as an equation
4. What do you notice?
You get the same solution as trial and error
x 3 3
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations

• Method 1 Common Base Method

Example 2 Solve for x 2x2 4x
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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 2x2 4x Does LSRS?
0
1
2
3
x 2x2 4x Does LSRS?
0 4 1 NO
1 8 4 NO
2 16 16 YES
3 32 64 NO
Work out the table then click ANSWER to see the
answer
ANSWER
The solution is x 2 Since, 222241642
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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 left side as a power with base 2
3. Solve for x
2x2(22)x 22x
x 2
2. Set the exponents as an equation
4. What do you notice?
You get the same solution as trial and error
x 2 2x
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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
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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 22x1 1
22x1 20
4. Set your exponents equal and solve
2. Write all powers with the same base
2x 1 0 2x -1 x -1/2
22x1 20
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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 LSRS. Below is
a graphical check of the solution
y2(22x)
x1/2
y1
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations

• Method 1 Common Base Method

Example 4 Solve 27x(92x-1) 3x4
33x(32)2x-1 3x4
33x(34x-2) 3x4
37x-2 3x4
7x 2 x 4 x 1
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations

• Method 1 Common Base Method

Example 5 Solve 4x3 4x 1040
4x(43) 4x 1040
64(4x) 4x 1040
65(4x) 1040
(4x) 16
(4x) 42 .x 2
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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.
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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.
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations
Method 2 Solving with Logarithms
Example 6 Solve for x 5x 27
log5x 27
Remember, whatever is done on one side must be
done to the other
log5x log27
log5x log27 xlog5 log27
Use the Laws of Logarithms to bring down the
exponent
Solve the equation by isolating the variable
log5x log27 xlog5 log27 log5 log 5
Use your calculator to determine log25/log5
log5x log27 xlog5 log27 log5 log 5 x
2.048
Try another example
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations
Method 2 Solving with Logarithms
Example 7 Solve for x 4x 8(x3)
log24x log28(x3)
Set the log to both sides. Use a base of 2 for
both sides.
log24x log28(x3) xlog24 (x3)log28
log24x log28(x3) xlog24 (x3)log28 xlog222
(x3)log223
Use the Power law of logarithms to bring down the
exponents
Write 4 and 8 as powers of base 2
Using the power property of logarithms the
following occurs Logbbmm
log24x log28(x3) xlog24 (x3)log28 xlog222
(x3)log223 x(2) (x3)(3)
Solve the equation by isolating the variable
log24x log28(x3) xlog24 (x3)log28 xlog222
(x3)log223 2x 3(x3)
log24x log28(x3) xlog24 (x3)log28 xlog222
(x3)log223 2x 3(x3) 2x 3x9
log24x log28(x3) xlog24 (x3)log28 xlog222
(x3)log223 2x 3(x3) x -9
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Unit 3 Exponential and Logarithmic Functions
Activity 6 Solving Exponential Equations
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