Name: Date:
Solve each exponential equation for x by expressing both sides in a common base, factoring, or using quadratic substitution. Check for extraneous roots.
Solve: 5^(2x - 1) = 125
Solve: 4^(x + 1) = 8^(x - 1)
Solve: (1/3)^x = 27^(x + 2)
Solve: 2^(x + 3) + 2^x = 36
Solve: 9^x - 4 * 3^x + 3 = 0
Solve: 4^x - 5 * 2^x - 24 = 0
Solve: 5^(2x - 1) = 125
x = 2. Rewrite 125 as 5^3, equate exponents 2x - 1 = 3, so 2x = 4.
Solve: 4^(x + 1) = 8^(x - 1)
x = 5. Rewrite with base 2: 2^(2(x + 1)) = 2^(3(x - 1)), so 2x + 2 = 3x - 3.
Solve: (1/3)^x = 27^(x + 2)
x = -1.5 (or -3/2). Rewrite with base 3: 3^(-x) = 3^(3(x + 2)), so -x = 3x + 6.
Solve: 2^(x + 3) + 2^x = 36
x = 2. Factor out 2^x to get 2^x(2^3 + 1) = 36, then 2^x(9) = 36, so 2^x = 4.
Solve: 9^x - 4 * 3^x + 3 = 0
x = 0 or x = 1. Let u = 3^x, so u^2 - 4u + 3 = 0, giving u = 1 or u = 3.
Solve: 4^x - 5 * 2^x - 24 = 0
x = 3. Let u = 2^x, so u^2 - 5u - 24 = (u - 8)(u + 3) = 0. Reject u = -3; solve 2^x = 8.
