Notes
Exponential models
| Model | Formula |
|---|---|
| Growth or decay | $y = a(1 \pm r)^t$ |
| Half-life $h$ | $\displaystyle y = a\left(\frac{1}{2}\right)^{t/h}$ |
| Compound interest | $\displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}$ |
| Continuous | $A = Pe^{rt}$ |
- $b > 1$ is growth and $0 < b < 1$ is decay. The number $e \approx 2.718$.
Logarithms
- $\log_b x = y$ means $b^y = x$. A log is an exponent.
- $\log x$ is base 10 and $\ln x$ is base $e$. Only positive numbers have logs.
- Properties: $\log_b MN = \log_b M + \log_b N$, $\displaystyle \log_b \frac{M}{N} = \log_b M - \log_b N$, $\log_b M^k = k\log_b M$.
- Change of base: $\displaystyle \log_b x = \frac{\ln x}{\ln b}$.
Solving
- Exponential: use a common base, or isolate the power and take logs.
- Logarithmic: combine into one log, rewrite as an exponential, and check that every log input is positive.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$24000(0.88)^3 \approx 16355.33$. Subtracting 36% once gives $15,360, which ignores compounding.
$\log_2 64 = 6$ and $\log_3 9 = 2$.
$\log_6 (4 \cdot 9) = \log_6 36 = 2$.
Divide by 3 first: $e^{2x} = 5$, so $\displaystyle x = \frac{\ln 5}{2} \approx 0.805$.
$\displaystyle \log_2 \frac{x + 6}{x} = 2 \Rightarrow x + 6 = 4x \Rightarrow x = 2$.
$\displaystyle 2 = e^{0.05t} \Rightarrow t = \frac{\ln 2}{0.05} \approx 13.86$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A lab culture starts with 300 bacteria and doubles every 4 hours, so $N(t) = 300 \cdot 2^{t/4}$, where $t$ is in hours.
(a) How many bacteria are there after 12 hours?
Show the answer and scoring guide
$N(12) = 300 \cdot 2^{3} = 2400$ bacteria.
Check off the points you earned:
(b) When will there be 9,600 bacteria? Solve without logarithms.
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$\displaystyle 300 \cdot 2^{t/4} = 9600 \Rightarrow 2^{t/4} = 32 = 2^5 \Rightarrow \frac{t}{4} = 5 \Rightarrow t = 20$ hours.
Check off the points you earned:
(c) When will there be 5,000 bacteria? Round to the nearest tenth of an hour.
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$\displaystyle 2^{t/4} = \frac{5000}{300} = \frac{50}{3}$. Take logs: $\displaystyle \frac{t}{4} = \frac{\ln(50/3)}{\ln 2} \approx 4.059$, so $t \approx 16.2$ hours.
Check off the points you earned: