Example 1: a simple common base

• Solve 2^{x + 1} = 32.
• Solution 32 = 2⁵, so 2^{x + 1} = 2⁵. Equating the exponents gives x + 1 = 5, so x = 4.

Example 2: two different bases that share a common base

• Solve 9^x = 27.
• Solution 9 = 3² and 27 = 3³. So 3^2x = 3³, which gives 2x = 3 and x = 3/2.

Example 3: a base that needs a linear exponent

Example 4: taking out a common factor

• Solve 3^{x + 2} + 3^x = 90.
• Solution Write 3^{x + 2} = 9 × 3^x. The equation becomes 9 × 3^x + 3^x = 90, that is 10 × 3^x = 90. So 3^x = 9 = 3², and x = 2.

Example 5: a quadratic in an exponential function

Example 6: a more demanding quadratic

• Solve 2^2x − 5 × 2^x + 4 = 0.
• Solution Let y = 2^x. Then y² − 5y + 4 = 0, so (y − 1)(y − 4) = 0, and y = 1 or y = 4.
• If 2^x = 1 then x = 0. If 2^x = 4 then x = 2. The solution set is {0, 2}.

Example 7: an equation with no solution

• Solve 2^x × 4^{x + 1} = 8^{x − 1}.
• Solution Write every side with base 2: the left side is 2^x × 2^{2x + 2} = 2^{3x + 2}, and the right side is 2^{3(x − 1)} = 2^{3x − 3}. Equating the exponents gives 3x + 2 = 3x − 3, that is 2 = −3. This is false for every x, so the equation has no solution.