M151Q Precalculus · Unit 2: Powers, exponentials, logarithms · on Exam 2
MFG 4.5 Exponential models
Study notes for Exponential models in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Doubling time and half-life characterize exponential growth and decay: $A(t) = A_0\,2^{t/D}$ or $A(t) = A_0\left(\tfrac12\right)^{t/H}$.
- Convert between forms: $2^{t/D} = b^t$ with $b = 2^{1/D}$; conversely $D = \dfrac{\log 2}{\log b}$.
- Fit an exponential to two data points (divide to find $b$), then answer questions with logs: "when does it reach 10,000?" means solve $A_0b^t = 10000$.
- Compound interest: $A = P(1 + r/n)^{nt}$; continuous compounding $A = Pe^{rt}$ (§5.3).
Key formulas
Doubling / half-life
$$A = A_0\,2^{t/D},\qquad A = A_0\left(\tfrac12\right)^{t/H},\qquad D = \frac{\log 2}{\log b},\quad H = \frac{\log 2}{\log(1/b)}$$
Solving for time
$$A_0b^t = A \Rightarrow t = \frac{\log(A/A_0)}{\log b}$$
Worked example
A drug's half-life is 6 hours. Starting at 200 mg, when is 25 mg left? What is the hourly decay factor?
Show the solution
$200(\tfrac12)^{t/6} = 25 \Rightarrow (\tfrac12)^{t/6} = \tfrac18 \Rightarrow t/6 = 3 \Rightarrow t = 18$ h. Hourly factor $b = (1/2)^{1/6} \approx 0.891$ (about 10.9% lost per hour).
Common mistakes
- Dividing the half-life by 2 to get a "quarter-life" (it takes two half-lives).
- Forgetting to isolate the exponential before taking a log.
Exam tipHalf-life questions often need no calculator: count the halvings. 200 → 100 → 50 → 25 is three half-lives.
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