Introduction
As we have already seen, exponential equations are
equations that have the variable in the exponent. Some
exponential equations are complex and some are quite
simple. In this lesson, we will focus on solving
exponential equations of the form b x = c, where b is the
base and x is the exponent.
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2.1.4: Solving Exponential Equations
Key Concepts
• It may help to look at the laws of exponents. These
laws, sometimes referred to as properties, are the
rules that must be followed when working with
exponents. The following table summarizes these
laws.
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2.1.4: Solving Exponential Equations
Key Concepts, continued
Laws of Exponents
Law
Multiplication of
exponents
General rule
b m • b n = b m + n
46 • 43 = 49
(b m) n = b mn
Power of exponents
(bc) n = b cc n
Division of
exponents
bm
Exponents of zero
b 0 = 1
bn
=b
Negative exponents b-n =
(4 6 ) 3 = 4 18
(4 • 2) 3 = 4 32 3
46
m-n
1
bn
Specific example
= 4 6 - 3 = 43
43
40 = 1
and
1
b-n
= bn
4-3 =
1
43
and
1
4-3
= 43
3
2.1.4: Solving Exponential Equations
Key Concepts, continued
• Keep these laws in mind when solving exponential
equations.
• There are two forms of exponential equations. One
form is used when each side of the equation can be
written using the same base, such as ab = ac. In this
case, b and c must be equal as long as a > 0 and a ≠ 1.
• The second form of exponential equations is used
when it isn’t possible to write each side of the
equation using the same base. How to solve this type
of exponential equation will be covered in a later
lesson.
2.1.4: Solving Exponential Equations
4
Key Concepts, continued
•
Follow a few basic guidelines to solve an exponential
equation where the bases of both sides of the
equation can be written so that they are equal.
Solving Exponential Equations
1. Rewrite the bases as powers of a common base.
2. Substitute the rewritten bases into the original
equation.
3. Simplify exponents.
4. Solve for the variable.
5
2.1.4: Solving Exponential Equations
Common Errors/Misconceptions
• attempting to solve an exponential equation as if it is a
linear equation
• not finding a common base prior to attempting to
solve the equation
• misidentifying the common base
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2.1.4: Solving Exponential Equations
Guided Practice
Example 1
Solve 4x = 1024.
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2.1.4: Solving Exponential Equations
Guided Practice: Example 1, continued
1. Rewrite the base as powers of a common base.
You may not recognize right away if it is possible to
write 1,024 as an exponential expression with a base
of 4. Begin by finding values of powers of 4 to see if it
is possible.
41 = 4
44 = 256
42 = 16
45 = 1024
43 = 64
We now know that it is possible to write 1,024 as a
power of 4.
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2.1.4: Solving Exponential Equations
Guided Practice: Example 1, continued
2. Rewrite the equation so that both sides
have a base of 4.
4x = 45
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2.1.4: Solving Exponential Equations
Guided Practice: Example 1, continued
3. Now solve for x by setting the exponents
equal to each other.
x=5
The solution to the equation 4x = 1024 is x = 5.
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2.1.4: Solving Exponential Equations
Guided Practice: Example 1, continued
4. Check your answer.
Substitute 5 for the variable x in the original equation.
4x = 1024
45 = 1024
1024 = 1024
This is a true statement.
✔
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2.1.4: Solving Exponential Equations
Guided Practice: Example 1, continued
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2.1.4: Solving Exponential Equations
Guided Practice
Example 4
Solve the equation 117 = 5 x – 8.
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2.1.4: Solving Exponential Equations
Guided Practice: Example 4, continued
1. Begin by eliminating the subtraction of 8
from the right side of the equal sign.
Do so by adding 8 to the equation.
117 = 5 x - 8
+8
+8
125 = 5 x
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2.1.4: Solving Exponential Equations
Guided Practice: Example 4, continued
2. Rewrite the base as powers of a common
base.
125 can be written as 5 to the power of 3.
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2.1.4: Solving Exponential Equations
Guided Practice: Example 4, continued
3. Rewrite the equation so both sides have a
base of 5.
125 = 5x
53 = 5x
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2.1.4: Solving Exponential Equations
Guided Practice: Example 4, continued
4. Now solve for x by setting the exponents
equal to each other.
x=3
The solution to the equation 117 = 5x – 8 is x = 3.
17
2.1.4: Solving Exponential Equations
Guided Practice: Example 4, continued
5. Check your answer.
Substitute 3 for the variable x in the original equation.
117 = 5x – 8
117 = 53 – 8
117 = 125 – 8
117 = 117
This is a true statement.
✔
2.1.4: Solving Exponential Equations
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Guided Practice: Example 4, continued
19
2.1.4: Solving Exponential Equations