Download Pre-Calculus 12A Section 7.3 Solving Exponential Equations

Pre-Calculus 12A
Section 7.3
Solving Exponential Equations
Exponential equation: An equation that has a variable in an exponent.
Example 1: Rewriting Powers with a Specified Base
Rewrite each of the following powers with a base of 2.
1
a. 32
b. 16
3
c.
3
16
 1 3
d.  
 64 
Solution:
1
a. 32 =
b. 16 =
c.
3
3
16 =
 1 3
d.   =
 64 
Example 2: Solve an Exponential Equation by Using a Common Base
Solve each equation.
x2
a.
252x 4  125x 1
d. 3
b.
123x 9  1
e.
c.
x2
8
1
 
4
 27  32x 
 3
3x
 92 x5
x 3
Solution:
a. Express both sides of the
equation as powers with base
_____:
252x 4  125x 1
Since both sides are single powers
with the same base, the exponents
must be equal.
Equate the exponents and solve:
b. Express both sides of the
equation as powers with base
_____:
123x 9  1
c. Express both sides of the
equation as powers with base
_____:
8
x2
1
 
4
x 3
Equate the exponents and solve:
Equate the exponents and solve:
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Pre-Calculus 12A
Section 7.3
d. Express both sides of the
equation as powers with base
_____:
3x  27  32x 
2
Since both sides are single powers
with the same base, the exponents
must be equal.
e. Express both sides of the
equation as powers with base
_____:
 3
3x
 92 x5
Equate the exponents and solve:
Equate the exponents and solve:
Example 3: Solving Problems Involving Exponential Equations with the Same Base
The half-life of radon-222 is 92 hours. If the initial amount of radon-222 was 48g, how long would it take for
the radon to decay to 3 g?
Solution:
Find the equation of the function that describes the amount of radon present in terms of time.
Express both sides of the equation as powers with the same base. Equate the exponents and solve.
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Pre-Calculus 12A
Section 7.3
Example 4: Solve Problems Involving Exponential Equations
A painting doubles in value every 8 years. It is currently worth $1000.
a. Write an exponential function that models the value, V, of the painting after t years.
b. Use your equation to determine the value of the painting in 12 years.
c. Use your equation to determine the time needed for the painting to be worth $32 000.
Solution:
a. The exponential function can be written in the form y  acbx :
b. Substitute t = 12 into your equation and solve for V:
c. Substitute V = 32 000 into your equation and solve for t:
Example 5: Solve Problems Involving Exponential Equations with Different Bases
The student council of a school notices that their membership is growing by 3% per year.
a. The membership is currently 350 students. Write an exponential function to model the size of the
student council. Define your variables.
b. Use your equation to determine the time needed until the student council has 560 members. Round
your answer to the nearest whole number.
Solution:
a. The exponential function can be written in the form y  ac x :
b. Substitute N = 560 into your equation and simplify.
We cannot express both sides of this equation with the same base. Use systematic trial to find the
approximate value of t that satisfies the equation.
The student council will have 560 members in approximately __________ years.
Note: We will learn how to solve equations like this algebraically when we study logarithms in the next unit.
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