Exponential and Logarithmic Functions
5.2 Exponential Functions
Exponential Growth and Decay
Exponential models describe repeated percent change.
Definition: Exponential Function
An exponential function has form
$$f(x)=ab^x,$$
where $a\ne0$, $b>0$, and $b\ne1$.
Definition: Growth and Decay
- If $b>1$, exponential growth.
- If $0<b<1$, exponential decay.
Example: Evaluate an exponential function
Let
$$f(x)=3\cdot2^x.$$
Find $f(0)$ and $f(4)$.
Show Solution
$$f(0)=3\cdot2^0=3,$$ $$f(4)=3\cdot2^4=48.$$
MyOpenMath: Practice: evaluating exponentials (placeholder)
Compound Interest Models
Many financial formulas are exponential in time.
Theorem: Compound interest
For principal $P$, annual rate $r$, compounded $n$ times per year for $t$ years:
$$A=P\left(1+\frac{r}{n}\right)^{nt}. $$
Example: Compute compound amount
Invest $2000 at 6% annual interest, compounded monthly for 3 years.
Show Solution
Use $P=2000$, $r=0.06$, $n=12$, $t=3$:
$$A=2000\left(1+\frac{0.06}{12}\right)^{36}=2000(1.005)^{36}. $$
Approximate:
$$A\approx 2393.70.$$