Exponential Functions

The Exponential Function

Every function $f(x) = a^x$ passes through $(0, 1)$, because $a^0 = 1$, and through $(1, a)$, because $a^1 = a$. Drag the slider and watch those two points stay put.

One base is special: the irrational number $e \approx 2.71828\ldots$. It shows up in continuous growth, such as interest compounded continuously, radioactive decay, and cooling. The function $e^x$ is the "default" exponential in science and calculus. Leave $e$ as $e$ unless a problem asks for a decimal.

Finding the equation from two points

If you know that $f(x) = k \cdot a^x$ passes through two points, you have two unknowns, $k$ and $a$. If one point has $x = 0$, then $a^0 = 1$ and $k$ is just the $y$-value. Otherwise, divide the two equations so that $k$ cancels.

Growth and Decay

Matching graphs to equations

When you must match a graph to its equation, check these clues in order:

1. Is it rising or falling? (base above or below $1$)

2. Where does it cross the $y$-axis? $a^x$ crosses at $(0, 1)$. $-a^x$ crosses at $(0, -1)$.

3. Which one is steeper for $x > 0$? Among bases larger than $1$, the larger base is steeper to the right of the $y$-axis.

The letters in this picture only label these four curves. In the exercise, the letters may sit on different curves, so use the clues above to match each graph.

Transformations, Asymptotes, and Intercepts

The graph of $a^x$ has the $x$-axis as a horizontal asymptote: the curve gets as close as you like to the line $y = 0$ but never touches it. When you shift the graph, the asymptote moves with it.

Intercepts. For the $y$-intercept, compute $f(0)$. For the $x$-intercept, set $f(x) = 0$ and solve.

Equations with a Common Base

An exponential equation has the unknown in an exponent. The simplest idea: if both sides are powers of the same base, the exponents must be equal.

Strategy: rewrite every number as a power of a common base (for example $8 = 2^3$, $\tfrac14 = 2^{-2}$, $125 = 5^3$), use the power rules, then set the exponents equal.

Factoring out a common power

Sometimes the terms have the same base but different exponents. Split each power with $a^{x+m} = a^x \cdot a^m$, then factor out $a^x$.

The graph above shows why $3^x = 9$ has exactly one solution: the horizontal line $y = 9$ crosses the curve once, at $x = 2$.

Substitution, Systems, and Logarithms

Substitution

If an equation has both $a^{2x}$ and $a^x$, it is quadratic in form. Since $a^{2x} = (a^x)^2$, let $t = a^x$. You get a quadratic in $t$ (review factoring or the quadratic formula if needed). Solve it, then go back to $x$.

Systems

If a system has $a^x$ and $b^y$ in it, treat each power as a single unknown, solve the linear system, then solve for the exponents.

Logarithms

When the two sides cannot be written with the same base, as in $5^{x+1} = 20$, take a logarithm of both sides. We use the natural log $\ln$ (the log with base $e$). The key property: $$\ln\left(a^x\right) = x \ln a.$$ This moves the unknown out of the exponent.

Domain and Asymptotes of Expressions with Exponentials

Two facts make these problems easier:

• $e^{(\text{anything})} > 0$, so an exponential factor is never $0$.

• The domain of $a^{u(x)}$ is the domain of the exponent $u(x)$. A fraction in the exponent or in the denominator causes restrictions.

Domain. Exclude every $x$ that makes a denominator $0$. If the denominator has an exponential, solve the equation $a^x = c$.

Asymptotes of $e^{u(x)}$. Look at what the exponent $u(x)$ does:

• If $u(x) \to L$ as $x \to \pm\infty$, then $e^{u(x)} \to e^L$, so there is a horizontal asymptote $y = e^L$.

• If $u(x) \to +\infty$ near a restricted value $x = c$, then $e^{u} \to \infty$. If $u(x) \to -\infty$, then $e^u \to 0$. A vertical asymptote appears only when the exponent blows up to $+\infty$ (it can be on one side only).

Sign, Continuity, and Intersections

Sign of a product

Since $e^{x}$ (or $e^{-x}$, or $2^x$) is always positive, it never changes the sign of a product. Only the other factor decides.

Continuity of a piecewise function

A piecewise function is continuous if you can draw it without lifting your pencil. Each piece is continuous on its own, so check the boundaries: the pieces must have the same value where they meet.

Where two graphs intersect

Set the two expressions equal. If they are powers of the same base, equate the exponents. Then plug the $x$-value into either function to get the $y$-coordinate. A triangle's area is $\tfrac12 \cdot \text{base} \cdot \text{height}$. When two vertices lie on a horizontal line, the base is the horizontal distance between them, and the height is the vertical distance from the third vertex to that line.

Growth and Decay Models

Anything that is multiplied by the same factor in each equal time period is modeled by an exponential function.

Solving for Time: Depreciation, Logistic Growth, and Cooling

When the unknown is the time, it sits in an exponent. Isolate the power, then take $\ln$ of both sides.

Two other models use $e$ with a constant $k$:

• Logistic growth (such as a rumor or an epidemic in a fixed population): $P(t) = \dfrac{L}{1 + Ae^{-kt}}$. Its starting value is $P(0) = \dfrac{L}{1 + A}$. As $t$ grows, $e^{-kt} \to 0$, so $P(t)$ approaches the limit $L$.

• Newton's law of cooling: $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is the room temperature and $T_0$ the starting temperature.

Back to the video: the views are multiplied by $2$ each hour, not increased by $100$. So $V(t) = 100 \cdot 2^t$, and after $10$ hours $V(10) = 100 \cdot 1024 = 102{,}400$ views. Adding $100$ per hour would give only $100 + 100 \cdot 10 = 1{,}100$. That difference is the power of an exponent.

Summary