HONORS ALGEBRA 2 : Problems on Solving Exponential Equations

Solve each of the following equations without using calculator.

Problem 1 :

5^x = 625

Solution :

5^x = 625

625 is a power of 5 and it can be written as 5⁴.

5^x = 5⁴

Since the base is same on both sides, exponents can be equated.

x = 4

Problem 2 :

Solution :

Problem 3 :

Solution :

Problem 4 :

3^{x - 10} = 27^x

Solution :

3^{x - 10} = 27^x

3^{x - 10} = (3³)^x

3^{x - 10} = 3^3x

x - 10 = 3x

2x = 10

x = 5

Problem 5 :

4^{x + 2} = 8^x

Solution :

4^{x + 2} = 8^x

(2²)^{x + 2} = (2³)^x

2^{2(x + 2)} = 2^3x

2(x + 2) = 3x

2x + 4 = 3x

4 = x

Problem 6 :

Solution :

x² + 1 = 2x

x² - 2x + 1 = 0

Solve by factoring.

x² - x - x + 1 = 0

x(x - 1) - 1(x - 1) = 0

(x - 1)(x - 1) = 0

(x - 1)² = 0

Taking square root on both sides,

x - 1 = 0

x = 1

Problem 7 :

Solution :

2(x + 4) = 3x²

3x² - 2x - 8 = 0

Solve by factoring.

3x² - 6x + 4x - 8 = 0

3x(x - 2) + 4(x - 2) = 0

(x - 2)(3x + 4) = 0

x - 2 = 0  or  3x + 4 = 0

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