Logarithms Test Answer Key with Scoring – Precalculus

Question 1 Evaluate Logarithms (2 points)

1. 3⁻⁴ = 1/81, so −4.
2. 8^1/3 = 2, so 1/3.

Scoring: 1 point per value.

Question 2 Expand and Condense (2 points)

(a) log₅ 25 + 2 log₅ x − log₅ y^1/2 = 2 + 2 log₅ x − (1/2) log₅ y.

(b) ln 9 + ln 16 − ln 6 = ln(144/6) = ln 24.

Scoring: 1 point per part.

Question 3 Solve an Exponential Equation (2 points)

(2x − 1) ln 3 = ln 14, so 2x − 1 = ln 14 ÷ ln 3 ≈ 2.4022.

x = 1 + ln 14 ÷ ln 32 ≈ 1.701.

Scoring: 1 point for taking logs correctly; 1 point for the exact and decimal answer.

Question 4 Logarithmic Equation (3 points)

Combine: log₆[x(x − 5)] = 1, so x² − 5x = 6 and x² − 5x − 6 = (x − 6)(x + 1) = 0.

The candidate −1 makes log₆(−1) undefined. The candidate 6 gives log₆ 6 + log₆ 1 = 1 + 0 = 1 ✓. x = 6.

Scoring: 1 point for combining logs; 1 point for solving the quadratic; 1 point for rejecting −1.

Question 5 Graph of a Shifted Logarithm (2 points)

1. Asymptote x = 1; domain (1, ∞).
2. Rewrite log₂(x − 1) = 3 as x − 1 = 8, so x = 9.

Scoring: 1 point per part.

Question 6 Continuous Growth of a Town (3 points)

1. P(10) = 18500e^0.24 ≈ 23,518 people.
2. Solve e^0.024t = 2: t = ln 2 ÷ 0.024 ≈ 28.9 years.
3. e^0.024t = 30000/18500, so t = ln(1.6216) ÷ 0.024 ≈ 20.1 years.

Scoring: 1 point per part.

Question 7 Logistic Model of a Herd (2 points)

1. Capacity 900; N(0) = 900 ÷ 18 = 50.
2. 1 + 17e^−0.6t = 2, so e^−0.6t = 1/17 and t = ln 17 ÷ 0.6 ≈ 4.72 years.

Scoring: 1 point per part.

Question 8 Logarithmic Scales (3 points)

(a) 7.1 − 5.4 = 1.7, so the ratio is 10^1.7 ≈ 50.1 times.

(b) M = log 2000 ≈ 3.30.

(c) [H⁺] = 10^−3.4 ≈ 3.98 × 10⁻⁴ mol/L.

Scoring: 1 point per part.

Question 9 Find an Inverse (1 point)

Swap and solve: x = ln(y − 3) + 1, so ln(y − 3) = x − 1 and y = e^{x − 1} + 3.

f⁻¹(x) = e^{x − 1} + 3.

Scoring: 1 point for the correct inverse.