Training Exponentials & Logarithms Exponential Functions
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Exponential Functions

20 min Exponentials & Logarithms

Exponential Functions

Exponential functions model quantities that grow or decay by a constant percentage in each time period. Population growth, radioactive decay, compound interest, and the spread of a virus all follow exponential patterns — and the mathematics is the same in every case.

The general form f(x) = a · bˣ captures two parameters: the initial amount a and the base b. When b is greater than 1, the function models growth; when b is between 0 and 1, it models decay.

This lesson introduces exponential functions, their graphs, and their defining characteristic — a constant ratio between successive outputs rather than a constant difference.

Definition

An exponential function has the form:

$$f(x) = a \cdot b^x$$

where $a \ne 0$, $b > 0$, and $b \ne 1$. The constant $b$ is the base and $a$ is the initial value (the $y$-intercept when $x = 0$).

Key Features

Feature$b > 1$ (Growth)$0 < b < 1$ (Decay)
BehaviorIncreasingDecreasing
As $x \to \infty$$f(x) \to \infty$$f(x) \to 0$
As $x \to -\infty$$f(x) \to 0$$f(x) \to \infty$
Horizontal asymptote$y = 0$$y = 0$
$y$-intercept$(0, a)$$(0, a)$
Example 1

Evaluate $f(x) = 3 \cdot 2^x$ at $x = -2, 0, 1, 3$.

$x$$f(x) = 3 \cdot 2^x$
$-2$$3 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = \frac{3}{4}$
$0$$3 \cdot 1 = 3$
$1$$3 \cdot 2 = 6$
$3$$3 \cdot 8 = 24$

The Number $e$

Definition

The number $e \approx 2.71828\ldots$ is called Euler's number. It is the base of the natural exponential function $f(x) = e^x$. It arises naturally in continuous growth/decay.

Compound Interest

Formulas

Periodic compounding: $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$

Continuous compounding: $A = Pe^{rt}$

where $P$ = principal, $r$ = annual rate, $n$ = compounds per year, $t$ = years.

Example 2

$\$5{,}000$ is invested at 6% compounded quarterly. Find the balance after 10 years.

$$A = 5000\left(1 + \frac{0.06}{4}\right)^{4 \times 10} = 5000(1.015)^{40} \approx \$9{,}070.09$$

Example 3

A bacteria culture doubles every 3 hours, starting with 200 bacteria. Write a function and find the count after 12 hours.

$f(t) = 200 \cdot 2^{t/3}$.

At $t=12$: $f(12) = 200 \cdot 2^4 = 200 \cdot 16 = 3{,}200$.

Example 4

A car worth $\$25{,}000$ depreciates 15% per year. Write the model and find its value after 5 years.

$V(t) = 25000(1 - 0.15)^t = 25000(0.85)^t$.

$V(5) = 25000(0.85)^5 \approx \$11{,}093$.

Practice Problems

1. Evaluate $f(x) = 5 \cdot 3^x$ at $x = 0, 2, -1$.
2. Is $f(x) = 4(0.7)^x$ growth or decay?
3. $\$2{,}000$ at 5% compounded monthly for 8 years. Find $A$.
4. A population of 500 grows at 12% per year. Write the function.
5. Evaluate $e^0$ and $e^1$ (approximate).
6. A radioactive substance has a half-life of 10 years. Starting with 80g, how much remains after 30 years?
7. $\$10{,}000$ at 4% compounded continuously for 5 years. Find $A$.
8. Evaluate $2^{-3}$ and $\left(\frac{1}{2}\right)^{-3}$.
9. A painting worth $\$800$ appreciates at 7% per year. Value after 20 years?
10. Compare: Which is larger, $3^{10}$ or $10^3$?
Show Answer Key

1. $f(0)=5$, $f(2)=45$, $f(-1)=\frac{5}{3}$

2. Decay (base $0.7 < 1$)

3. $A = 2000(1.004167)^{96} \approx \$2{,}983.52$

4. $P(t) = 500(1.12)^t$

5. $e^0 = 1$; $e^1 \approx 2.718$

6. $80 \cdot (0.5)^3 = 10$ g

7. $A = 10000e^{0.2} \approx \$12{,}214.03$

8. $2^{-3} = \frac{1}{8}$; $(\frac{1}{2})^{-3} = 8$

9. $800(1.07)^{20} \approx \$3{,}094.13$

10. $3^{10} = 59{,}049$ vs. $10^3 = 1{,}000$. So $3^{10}$ is much larger.

Introduction to Logarithms