SOLVING EXPONENTIAL EQUATIONS BY REWRITING THE BASE

Exponential equation is an equation whose exponent or part of the exponent is a variable.

Examples :

2^x = 8

7^{3x - 1} = 49

Solve for x in each of the following :

Example 1 :

2^x = 32

Solution :

2^x = 32

2^x = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2

2^x = 2⁵

Equate the exponents.

x = 5

Example 2 :

3^{x - 2} = 1/9

Solution :

3^{x - 2} = 1/9

3^{x - 2} = 1/3²

3^{x - 2} = 3^-2

Equate the exponents.

x - 2 = -2

Add 2 to both sides.

x = 0

Example 3 :

6 ⋅ 3^x = 54

Solution :

6 ⋅ 3^x = 54

Divide both sides by 6.

3^x = 54/6

3^x = 9

3^x = 3²

Equate the exponents.

x = 2

Example 4 :

4^{x - 1} = (1/2)^{1 - 3x}

Solution :

4^{x - 1} = (1/2)^{1 - 3x}

2^{2(x - 1)} = 2^{-1(1 - 3x)}

2^{2x - 2} = 2^{-1 + 3x}

Equate the exponents.

2x - 2 = -1 + 3x

Subtract 3x from both sides.

-x - 2 = -1

Add 2 to both sides.

-x = 1

Multiply both sides by -1.

x = -1

Example 5 :

2^{x + 2} = 1/4

Solution :

2^{x + 2} = 1/4

2^{x + 2} = 1/2²

2^{x + 2} = 2^-2

Equate the exponents.

x + 2 = -2

Subtract 2 from both sides.

x = -4

Example 6 :

3^{1 - 2x} = 1/27

Solution :

3^{1 - 2x} = 1/27

3^{1 - 2x} = 1/3³

3^{1 - 2x} = 3^-3

Equate the exponents.

1 - 2x = -3

Subtract 1 from both sides.

-2x = -4

Divide both sides by -2.

x = 2

Example 7 :

5 ⋅ 2^x = 40

Solution :

5 ⋅ 2^x = 40

Divide both sides by 5.

2^x = 40/5

2^x = 8

2^x = 2³

Equate the exponents.

x = 3

Example 8 :

9^x = 7(3^x) + 18

Solution :

9^x = 7(3^x) + 18

(3²)^x = 7(3^x) + 18

(3^x)² = 7(3^x) + 18

(3^x)² - 7(3^x) - 18 = 0

Let a = 3^x.

a² - 7a - 18 = 0

(a - 9)(a + 2) = 0

a - 9 = 0  or  a + 2 = 0

In 3^x, whatever real value (positive or negative or zero) we substitute for x, 3^x can never be negative. So we can ignore the equation 3^x = -2.

Therefore,

x = 2

Example 9 :

3^{x + 1} ⋅ 9^-x = (1/3)^{x + 1}

Solution :

3^{x + 1} ⋅ 9^-x = (1/3)^{x + 1}

3^{x + 1} ⋅ (3²)^-x = (3^-1)^{x + 1}

3^{x + 1} ⋅ 3^-2x = 3^{-1(x + 1)}

3^{x + 1 - 2x} = 3^{-x - 1}

3^{-x + 1} = 3^{-x - 1}

Equate the exponents.

-x + 1 = -x - 1

Add x to both sides.

1 = -1

In the final step solving the given equation, the variable is no more. And also, 1 = -1 is a false statement, so, there is no solution for this equation.

Example 10 :

2^x ⋅ 4^{2 - x} = 8

Solution :

2^x ⋅ 4^{2 - x} = 8

2⋅ (2²)^{2 - x} = 2³

2¹⋅ 2^{4 - 2x} = 2³

2^{1 + 4 - 2x} = 2³

2^{5 - 2x} = 2³

Equate the exponents.

5 - 2x = 3

Subtract 5 from both sides.

-2x = -2

Divide both sides by -2.

x = 1

Example 11 :

2^{x - 1}⋅ 4^{2x + 1} = 32

Solution :

2^{x  -1}⋅ 4^{2x + 1} = 32

2^{x - 1}⋅ (2²)^{2x + 1} = 2⁵

2^{x - 1}⋅ 2^{2(2x + 1)} = 2⁵

2^{x - 1}⋅ 2^{4x + 2} = 2⁵

2^{x - 1 + 4x + 2} = 2⁵

2^{5x + 1} = 2⁵

Equate the exponents.

5x + 1 = 5

Subtract 1 from both sides.

5x = 4

Divide both sides by 5.

x = 4/5

Example 12 :

5^{3x - 2} = 125^2x

Solution :

5^{3x - 2} = 125^2x

5^{3x - 2} = (5³)^2x

5^{3x - 2} = 5^6x

Equate the exponents.

3x - 2 = 6x

Subtract 6x from both sides.

-3x - 2 = 0

Add 2 to both sides.

-3x = 2

Divide both sides by -3.

x = -2/3

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