Notes
Exponential functions
- $f(x) = a \cdot b^x$: growth if $b > 1$, decay if $0 < b < 1$. Growth rate $r$ gives $b = 1 + r$; decay rate $r$ gives $b = 1 - r$.
- Compound interest: $\displaystyle A = P(1 + \frac{r}{n})^{nt}$. Continuous: $A = Pe^{rt}$.
Logarithms
- $\log_b x = y$ means $b^y = x$. Logs and exponentials are inverses.
- Properties: $\log_b (MN) = \log_b M + \log_b N$, $\displaystyle \log_b \frac{M}{N} = \log_b M - \log_b N$, $\log_b M^p = p \log_b M$.
- Change of base: $\displaystyle \log_b x = \frac{\ln x}{\ln b}$.
Solving equations
- Match bases when possible. Otherwise isolate the power and take a log of both sides.
- For logs, combine, rewrite in exponential form, then check that every log input is positive.
Topics in this unit
Each topic has its own short notes and practice.
Practice
A 3% increase each year multiplies by $1.03$.
$3^4 = 81$.
The power property gives $\log x^3$, and the quotient property turns the subtraction into division.
$\displaystyle \frac{1}{8} = 2^{-3}$.
$2x - 1 = 5^2 = 25$, so $x = 13$.
$200(0.90)^2 = 200(0.81) = 162$. The decreases compound, so it is not $200 - 2(20)$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Maya invests $5,000 in an account that earns 4% interest compounded annually.
(a) Write a function for the balance after $t$ years and find the balance after 5 years.
Show the answer and scoring guide
$A(t) = 5000(1.04)^t$. $A(5) = 5000(1.04)^5 \approx 5000(1.2167) = $6,083.26$.
Check off the points you earned:
(b) Write and solve an equation for the number of years until the balance reaches $8,000.
Show the answer and scoring guide
$\displaystyle 5000(1.04)^t = 8000 \Rightarrow 1.04^t = 1.6 \Rightarrow t = \frac{\ln 1.6}{\ln 1.04} \approx 11.98$. It takes about 12 years.
Check off the points you earned:
(c) How long does it take for the money to double?
Show the answer and scoring guide
$\displaystyle 1.04^t = 2 \Rightarrow t = \frac{\ln 2}{\ln 1.04} \approx 17.67$ years, about 17.7 years.
Check off the points you earned: