Worked examples · Sheet 1

Exponential equations with the same base

Theory used: Equality of powers with the same base Rules of powers

Exercise 1

Solve the equation

3^{x} = 81.

3^{x} = 81 ⟺ 3^{x} = 3^{4} ⟺ x = 4

Solution: $x = 4$ .

The number $81$ is a power of $3$ , since $3^{4} = 81$ , so the two sides can be written as powers of the same base. The exponential function with base $3$ is injective, therefore the equality of the powers is equivalent to the equality of the exponents, that is, $x = 4$ .

Exercise 2

Solve the equation

2^{x + 1} = 8^{x - 1} .

8^{x - 1} = (2^{3})^{x - 1} = 2^{3 x - 3}

2^{x + 1} = 2^{3 x - 3} ⟺ x + 1 = 3 x - 3 ⟺ 4 = 2 x ⟺ x = 2

Solution: $x = 2$ .

The bases $2$ and $8$ are different, but $8 = 2^{3}$ , so the right-hand side is rewritten as a power of $2$ with the power-of-a-power rule, which multiplies the exponents. The two sides are now powers of the same base, therefore the equation is equivalent to the equality of the exponents $x + 1 = 3 x - 3$ , a first-degree equation with solution $x = 2$ . The check confirms the result: for $x = 2$ the left-hand side equals $2^{3} = 8$ and the right-hand side equals $8^{1} = 8$ .

Exercise 3

Solve the equation

(\frac{1}{9})^{x} = 2 7^{x - 1} .

(\frac{1}{9})^{x} = (3^{- 2})^{x} = 3^{- 2 x}, 2 7^{x - 1} = (3^{3})^{x - 1} = 3^{3 x - 3}

3^{- 2 x} = 3^{3 x - 3} ⟺ - 2 x = 3 x - 3 ⟺ 3 = 5 x ⟺ x = \frac{3}{5}

Solution: $x = \frac{3}{5}$ .

Both $\frac{1}{9}$ and $27$ are powers of $3$ , since $\frac{1}{9} = 3^{- 2}$ and $27 = 3^{3}$ , so both sides are rewritten as powers with base $3$ using the power-of-a-power rule. The equality of the powers is then equivalent to the equality of the exponents $- 2 x = 3 x - 3$ ; moving the terms containing the unknown to the same side gives $5 x = 3$ , so $x = \frac{3}{5}$ .