Introduction As we have already seen, exponential equations are equations that have the variable in the exponent. Some exponential equations are complex

IntroductionAs 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.

2.1.4: Solving Exponential Equations

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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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Key Concepts, continued

Laws of Exponents

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Law General rule Specific exampleMultiplication of exponents

b m • b n = b m + n 46 • 43 = 49

Power of exponents(b m) n = b mn

(bc) n = b cc n(4 6 ) 3 = 4 18

(4 • 2) 3 = 4 32 3

Division of exponents

Exponents of zero b 0 = 1 40 = 1

Negative exponents and and

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.

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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.

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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.


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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Guided Practice

Example 1

Solve 4x = 1024.

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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. 8



2.Rewrite the equation so that both sides have a base of 4.

4x = 45

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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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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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Guided Practice

Example 4Solve the equation 117 = 5 x – 8.

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1.Begin by eliminating the subtraction of 8 from the right side of the equal sign.

Do so by adding 8 to the equation.

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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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3.Rewrite the equation so both sides have a base of 5.

125 = 5x

53 = 5x

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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.

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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.

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Documents

Introduction As we have already seen, exponential equations are equations that have the variable in the exponent. Some exponential equations are complex