Exponentials, Logs & Modelling

Practice Questions

Level 1: Easy (Foundational)

1. Determine $\log_{10}(100)$.
   A) $1$
   B) $2$
   C) $10$
   D) $100$
2. What is $3^4$?
   A) $12$
   B) $27$
   C) $64$
   D) $81$
3. A population of $200$ doubles every $5$ years. Determine the exact population after $10$ years. (SPR)

Level 2: Medium (Standard)

1. A radioactive sample decays with half-life $4$ years. Starting at $80$ g, determine the exact mass after $12$ years.
   A) $5$ g
   B) $10$ g
   C) $20$ g
   D) $40$ g
2. If $\log_2(x)=5$, determine $x$.
   A) $10$
   B) $16$
   C) $25$
   D) $32$
3. A bacteria colony triples every hour. Starting at $50$, determine the exact count after $3$ hours. (SPR)
4. A model $Q(t) = Q_0 e^{-kt}$ has half-life $\ln 2 / k$. If $k = 0.1$, determine, to the nearest year, the half-life.
   A) $3$
   B) $5$
   C) $7$
   D) $10$

Level 3: Hard (Advanced)

1. The equation $2 \cdot 3^{x} = 54$ has which exact solution?
   A) $x=2$
   B) $x=3$
   C) $x=\log_3 27$
   D) Both B and C
2. A quantity grows at continuous rate $r = 0.08$ per year. Determine the doubling time in years (nearest integer).
   A) $7$
   B) $9$
   C) $12$
   D) $14$
3. If $\ln x = 3$, determine, to $2$ decimal places, the value of $x$. (SPR)