Logarithms Test – Precalculus Math (Printable PDF)

Question 1 Evaluate Logarithms (2 points)

Evaluate without a calculator.

1. log₃(1/81) (1 point)
2. log₈ 2 (1 point)

Question 2 Expand and Condense (2 points)

(a) Expand log₅ 25x²√y. (1 point) (b) Write 2 ln 3 + 4 ln 2 − ln 6 as one logarithm. (1 point)

Question 3 Solve an Exponential Equation (2 points)

Solve 3^{2x − 1} = 14. Give the exact form and a decimal to three places.

Question 4 Logarithmic Equation (3 points)

Solve log₆ x + log₆(x − 5) = 1 and verify each candidate.

Question 5 Graph of a Shifted Logarithm (2 points)

The graph shows f(x) = log₂(x − 1) with a dashed vertical asymptote.

1. Give the equation of the asymptote and the domain. (1 point)
2. Solve f(x) = 3. (1 point)

Question 6 Continuous Growth of a Town (3 points)

A town of 18,500 people grows continuously at 2.4% per year, so P(t) = 18500e^0.024t.

1. Find the population after 10 years (nearest person). (1 point)
2. Find the doubling time to the nearest tenth of a year. (1 point)
3. When will the population reach 30,000? Round to the nearest tenth of a year. (1 point)

Question 7 Logistic Model of a Herd (2 points)

A herd on an island follows N(t) = 9001 + 17e^−0.6t (t in years), graphed below.

1. State the carrying capacity and the initial population. (1 point)
2. When does the herd reach 450 animals? Round to two decimals. (1 point)

Question 8 Logarithmic Scales (3 points)

(a) An earthquake of magnitude 7.1 and another of magnitude 5.4 are compared. How many times greater is the amplitude of the larger one? (1 point)

(b) A quake has amplitude 2,000 times the reference amplitude. Find its magnitude with M = log(A/A₀). (1 point)

(c) The pH of a solution is −log[H⁺]. Find the hydrogen-ion concentration when pH = 3.4. (1 point)

Question 9 Find an Inverse (1 point)

Find the inverse of f(x) = ln(x − 3) + 1.