6.1 Exponential Functions

Exponential Functions

Exponential functions model situations where a quantity grows or shrinks by a constant percentage over equal time intervals. That multiplicative behavior is what makes them different from linear functions, which grow by a constant amount. You'll see them everywhere: compound interest, population models, radioactive decay, and more.

Exponential Functions with Various Bases

An exponential function has the form $f(x) = b^x$, where $b$ is a positive constant and $b \neq 1$. The base $b$ determines the function's behavior:

• $b > 1$: The function increases as $x$ increases. This is exponential growth. For example, $f(x) = 2^x$ doubles every time $x$ increases by 1.
• $0 < b < 1$: The function decreases as $x$ increases. This is exponential decay. For example, $f(x) = (0.5)^x$ cuts in half every time $x$ increases by 1.

The natural exponential function $f(x) = e^x$ uses the special base $e \approx 2.71828$. This constant shows up naturally in continuous growth and decay models, and it's the default base for most scientific applications.

Solving exponential equations follows a consistent process:

1. Isolate the exponential expression on one side of the equation.
2. Take the logarithm of both sides (use $\ln$ if the base is $e$, or $\log$ for base 10, or $\log_b$ to match the base).
3. Use logarithm properties to bring the variable out of the exponent.
4. Solve for the variable.

For example, to solve $3^x = 20$: take $\log$ of both sides to get $x \cdot \log(3) = \log(20)$, then $x = \frac{\log(20)}{\log(3)} \approx 2.727$.

Features of Exponential Graphs

Every exponential function $f(x) = b^x$ shares a set of core graph features:

• Domain: All real numbers. You can plug in any value of $x$.
• Range: All positive real numbers, $(0, \infty)$. The output is never zero or negative.
• Horizontal asymptote at $y = 0$. The graph gets closer and closer to the x-axis but never touches it.
• y-intercept at $(0, 1)$, because $b^0 = 1$ for any valid base.
• The graph is increasing when $b > 1$ and decreasing when $0 < b < 1$.
• The curve is always concave up, meaning it bends upward regardless of whether it's growing or decaying.

Transformations shift and stretch the basic graph:

• Vertical shift: $f(x) = b^x + k$ moves the graph up ($k > 0$) or down ($k < 0$). This also moves the asymptote to $y = k$.
• Horizontal shift: $f(x) = b^{x-h}$ moves the graph right by $h$ units (left if $h$ is negative).
• Vertical stretch/compression: $f(x) = a \cdot b^x$ stretches the graph vertically by a factor of $a$. The y-intercept becomes $(0, a)$ instead of $(0, 1)$.
• Reflection: $f(x) = b^{-x}$ reflects the graph across the y-axis, turning growth into decay and vice versa.

A common mistake: students forget that a vertical shift changes the asymptote. If $f(x) = 2^x + 3$, the asymptote is $y = 3$, not $y = 0$.

Modeling with Exponential Equations

Compound interest is one of the most common applications. The formula is:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

• $P$ = principal (initial investment)
• $r$ = annual interest rate as a decimal (so 5% becomes 0.05)
• $n$ = number of times interest compounds per year (monthly = 12, quarterly = 4)
• $t$ = time in years
• $A$ = final amount

For example, if you invest $1,000 at 6% compounded monthly for 5 years:

$A = 1000\left(1 + \frac{0.06}{12}\right)^{12 \cdot 5} = 1000(1.005)^{60} \approx \text{\textdollar}1,348.85$

Continuously compounded interest uses the formula $A = Pe^{rt}$. This represents the limiting case where interest compounds an infinite number of times per year. With the same example at continuous compounding: $A = 1000 \cdot e^{0.06 \cdot 5} \approx \text{\textdollar}1,349.86$. Notice it's only slightly more than monthly compounding.

Population growth is modeled by $P(t) = P_0 e^{kt}$, where $P_0$ is the initial population and $k$ is the continuous growth rate. When $k > 0$, the population grows; when $k < 0$, it declines.

Radioactive decay follows $A(t) = A_0 e^{-\lambda t}$, where $\lambda$ is the decay constant (always positive, so the negative sign ensures the quantity decreases).

Key Formulas: Half-Life and Doubling Time

Two values come up constantly in application problems:

Doubling time is how long it takes a growing quantity to double. For a model with continuous growth rate $r$:

$t_{\text{double}} = \frac{\ln(2)}{r}$

Half-life is how long it takes a decaying quantity to drop to half its starting value. For a model with decay constant $\lambda$:

$t_{1/2} = \frac{\ln(2)}{\lambda}$

Both formulas come from the same idea: set the quantity equal to twice (or half) the starting amount and solve for $t$. The $\ln(2) \approx 0.693$ appears because you're solving $e^{rt} = 2$.

Applications of Exponential Growth and Decay

Bacterial growth: $N(t) = N_0 e^{rt}$, where $N_0$ is the initial bacteria count and $r$ depends on environmental factors like temperature and nutrients. If a colony of 500 bacteria grows at a rate of $r = 0.04$ per minute, after 2 hours (120 min): $N(120) = 500 \cdot e^{0.04 \cdot 120} = 500 \cdot e^{4.8} \approx 60,770$ bacteria.

Newton's Law of Cooling: $T(t) = T_a + (T_0 - T_a)e^{-kt}$

This models how an object's temperature approaches the surrounding (ambient) temperature $T_a$ over time. $T_0$ is the object's initial temperature, and $k$ is a cooling constant that depends on the material and environment. Notice that as $t \to \infty$, the exponential term drops to zero and $T(t) \to T_a$, which makes physical sense.

Logarithms and Inverse Functions

Logarithms reverse what exponential functions do. If $b^y = x$, then $\log_b(x) = y$. In plain terms, $\log_b(x)$ asks: "What exponent do you put on $b$ to get $x$?"

The two most common bases:

• Common logarithm (base 10): written $\log(x)$. Used in pH scales, decibels, and many applied settings.
• Natural logarithm (base $e$): written $\ln(x)$. Used whenever the model involves $e$, which is most continuous growth/decay problems.

Key facts about logarithmic functions:

• Domain: All positive real numbers, $(0, \infty)$. You cannot take the log of zero or a negative number.
• Range: All real numbers.
• The graph of $y = \log_b(x)$ is the reflection of $y = b^x$ across the line $y = x$, since they're inverse functions.

Logarithmic properties are the main tool for solving exponential equations, and you'll use them heavily in the sections ahead.