Exponential and Logarithmic Equations Test Answer Key with Scoring – Algebra 2

Question 1 Same-base equation (2 points)

81 = 3⁴, so x − 2 = 4 and x = 6.

Scoring: 1 point for writing 81 as 3⁴, 1 point for the solution.

Question 2 Exponential equation (2 points)

x = ln 50 ÷ ln 6 = 3.9120 ÷ 1.7918 = 2.1833. x ≈ 2.18.

Scoring: 1 point for the logarithm setup, 1 point for the value.

Question 3 Isolate and solve with e (2 points)

e^0.5x = 7.5, so 0.5x = ln 7.5 ≈ 2.0149 and x ≈ 4.030. x ≈ 4.03.

Scoring: 1 point for isolating, 1 point for the value.

Question 4 Logarithmic equation (2 points)

2x + 3 = 7² = 49, so 2x = 46. x = 23.

Scoring: 1 point for exponential form, 1 point for the solution.

Question 5 Equation with two logarithms (3 points)

log₅(x² + 4x) = 1, so x² + 4x = 5 and (x + 5)(x − 1) = 0. The value −5 makes log₅ x undefined. x = 1. Check: log₅ 1 + log₅ 5 = 0 + 1.

Scoring: 1 point for combining, 1 point for factoring, 1 point for rejecting −5.

Question 6 Decay model (3 points)

1. The curve is at 40 a little after 4 hours.
2. 100(0.8)^x = 40, so 0.8^x = 0.4 and x = ln 0.4 ÷ ln 0.8 = 4.106. ≈ 4.11 hours.

Scoring: 1 point for the estimate, 1 point for the logarithm setup, 1 point for the value.

Question 7 Compound interest (2 points)

2500(1 + 0.038/12)^12t = 4000, so (1.0031667)^12t = 1.6. Then 12t = ln 1.6 ÷ ln 1.0031667 = 0.4700 ÷ 0.00316, so t ≈ 12.388. ≈ 12.4 years.

Scoring: 1 point for the setup, 1 point for the value.

Question 8 Half-life (2 points)

200(1/2)^t/5 = 25, so (1/2)^t/5 = 1/8 = (1/2)³. Then t/5 = 3. t = 15 hours.

Scoring: 1 point for the equation, 1 point for the value.

Question 9 Extraneous solution (2 points)

x² + 9x = 10, so (x + 10)(x − 1) = 0. The value −10 is rejected because log(−10) is undefined. x = 1. Check: log 1 + log 10 = 0 + 1 = 1.

Scoring: 1 point for the solutions of the quadratic, 1 point for the rejection and answer.