Exponential Equations Worksheet

Exponential equations are algebraic equations where the variable is located in the exponent. They are typically in the form a^x = b, where a is a constant base and x is the exponent containing the variable.

Solved Examples

Example 1: Equation: 2^x = 16

Solution:

2^x = 2⁴

⇒ x = 4

Example 2: Equation: 3^x+1 = 81

Solution:

3^x+1 = 3⁴

⇒ x + 1 = 4

⇒ x = 3

Example 3: Equation: 5^2x = 125

Solution:

5^2x = 5³

⇒ 2x = 3

⇒ x = 3/2

Example 4: Equation: 4^x-1 = 64

Solution:

4^x-1 = 4³

⇒ x-1 = 3

⇒ x = 4

Example 5: Equation: 7^2x = 49

Solution:

7^2x = 7²

⇒ 2x = 2

⇒ x = 1

Example 6: Equation: 9^x+2 = 81

Solution:

9^x+2 = 9²

⇒ x + 2 = 2

⇒ x = 0

Example 7: Equation: 2^3x = 32

Solution:

2^3x = 2⁵

⇒ 3x = 5

⇒ x = 5/3

Example 8: Equation: 5^x+1 = 1/25

Solution:

5^x+1 = 5^-2

⇒ x + 1 = -2

⇒ x = -3

Example 9: Equation: 3^2x = 27^x-1

Solution:

3^2x = (3³)^x-1

⇒ 3^2x = 3^3(x-1)

⇒ 2x = 3(x-1)

⇒ 2x = 3x - 3

⇒ x = 3

Example 10: Solve for x: 2^{x^2 - 1} = 32

Solution:

2^{x^2 - 1} = 2^5

⇒ x² - 1 = 5

⇒ x² = 6

Thus, x = √6 or x = -√6

Example 11: Solve for x: e^2x = 7

Solution:

e^2x = 7

Take log both side, we get

ln e^2x = ln 7

⇒ 2x = ln 7

⇒ x = (ln 7)/2

Example 12: Solve 4^x = 2^2x+1

Solution:

(2^2)^x = 2^{2x+1}

\Rightarrow 2^{2x} = 2^{2x+1}

2x = 2x + 1

No solution

Practice Questions

Problem 1: Solve for x: 2^x = 16

Problem 2: Solve for x: 3^{2x} = 81

Problem 3: Solve for x: 5^{x+1} = 125

Problem 4: Solve for x: 4^x = 2^{2x+2}

Problem 5: Solve for x: 7^{x-2} = \frac{1}{49}

Problem 6: Solve for x: 9^{x+1} = 27

Problem 7: Solve for x: 10^{2x} = 1000

Problem 8: Solve for x: 6^{x+2} = 36^{x-1}

Problem 9: Solve for x: 8^{x-1} = 32

Problem 10: Solve for x: 2^{3x+1} = 16^{x-2}