Chapter 4: Exponential and Logarithmic Functions

Exponential and logarithmic functions are among the most important in all of mathematics. They model phenomena that grow or decay at rates proportional to their current value: population growth, radioactive decay, compound interest, sound intensity, and earthquake magnitude. In this chapter, we build up from the definition of exponential functions, introduce logarithms as their inverses, develop the algebraic properties that make logarithms so powerful, and then apply everything to solve real-world problems.

4.1 Exponential Functions

You have already encountered integer exponents: $2^3 = 8$, $5^{-1} = \frac{1}{5}$, and so on. An exponential function takes this idea and lets the exponent be any real number, creating a smooth, continuous curve that models growth and decay.

Why do we require $b > 0$? If $b$ were negative, then $b^x$ would be undefined for many values of $x$ (for example, $(-2)^{1/2}$ is not a real number). And why $b \neq 1$? Because $1^x = 1$ for all $x$, which is just a constant function and not very interesting.

Key Properties of $f(x) = b^x$

Every exponential function $f(x) = b^x$ shares several fundamental properties:

• Domain: All real numbers $(-\infty, \infty)$
• Range: All positive real numbers $(0, \infty)$. The output is always positive, no matter what $x$ is.
• $y$-intercept: The point $(0, 1)$, since $b^0 = 1$ for any valid base.
• Horizontal asymptote: The line $y = 0$ (the $x$-axis). The curve approaches but never touches or crosses this line.
• One-to-one: The function passes the horizontal line test, meaning it has an inverse function (which we will discover is the logarithm).

Growth vs. Decay

The behavior of $f(x) = b^x$ depends on whether the base is greater than or less than $1$:

• If $b > 1$, the function is increasing (exponential growth). As $x$ increases, $f(x)$ grows without bound. Example: $f(x) = 2^x$.
• If $0 < b < 1$, the function is decreasing (exponential decay). As $x$ increases, $f(x)$ shrinks toward zero. Example: $f(x) = \left(\frac{1}{2}\right)^x$.

Use the interactive graph below to explore how changing the base $b$ affects the shape of the exponential curve. When $b > 1$ the curve rises to the right; when $0 < b < 1$ it falls to the right. Notice that regardless of the base, every curve passes through $(0, 1)$.

Interactive: Drag the slider to change the base $b$ and observe growth vs. decay.

The Natural Exponential Function

Among all possible bases, one stands out as the most important in mathematics: the number $e$, known as Euler's number. It is an irrational constant defined by the limit

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828$$

The function $f(x) = e^x$ is called the natural exponential function. It arises naturally in calculus because it is the unique exponential function that is its own derivative: $\frac{d}{dx} e^x = e^x$. For this reason, $e^x$ appears throughout physics, engineering, finance, and biology whenever continuous growth or decay is modeled.

4.2 Logarithmic Functions

We have just seen that every exponential function $f(x) = b^x$ (with $b > 0, b \neq 1$) is one-to-one. This means it has an inverse function. That inverse is the logarithm.

Think of the logarithm as answering a question. The expression $\log_2(8)$ asks: "2 raised to what power gives 8?" Since $2^3 = 8$, the answer is $\log_2(8) = 3$.

Common and Natural Logarithms

Two bases are used so frequently that they have special notation:

• Common logarithm: $\log(x) = \log_{10}(x)$. When no base is written, base 10 is assumed. This is the "log" button on most calculators.
• Natural logarithm: $\ln(x) = \log_e(x)$. The inverse of the natural exponential $e^x$. This is the "ln" button on calculators.

Properties of $g(x) = \log_b(x)$

Since $g(x) = \log_b(x)$ is the inverse of $f(x) = b^x$, its properties mirror those of the exponential:

• Domain: $(0, \infty)$. You can only take the logarithm of a positive number.
• Range: All real numbers $(-\infty, \infty)$.
• $x$-intercept: The point $(1, 0)$, since $\log_b(1) = 0$ for any base.
• Vertical asymptote: The line $x = 0$ (the $y$-axis).
• Key point: $(b, 1)$ is always on the graph, since $\log_b(b) = 1$.

The Inverse Relationship

Because $f(x) = b^x$ and $g(x) = \log_b(x)$ are inverses, they "undo" each other:

$$b^{\log_b(x)} = x \quad \text{for } x > 0 \qquad \text{and} \qquad \log_b(b^x) = x \quad \text{for all } x$$

Graphically, the curves $y = b^x$ and $y = \log_b(x)$ are reflections of each other across the line $y = x$. The interactive graph below demonstrates this. Adjust the slider to change the base and observe how both curves change together while always remaining symmetric about the diagonal.

Interactive: The exponential (blue) and its inverse logarithm (red) reflected across $y = x$ (dashed).

4.3 Properties of Logarithms

Logarithms were originally invented to simplify computation: they turn multiplication into addition, division into subtraction, and exponentiation into multiplication. These properties arise directly from the corresponding exponent rules and remain essential tools for solving equations and simplifying expressions.

These properties follow from the exponent laws. For instance, the product rule works because if $\log_b(M) = p$ and $\log_b(N) = q$, then $b^p = M$ and $b^q = N$, so $MN = b^p \cdot b^q = b^{p+q}$, which means $\log_b(MN) = p + q = \log_b(M) + \log_b(N)$.

4.4 Solving Exponential and Logarithmic Equations

The key to solving exponential equations is to isolate the exponential expression and then apply a logarithm to both sides. The key to solving logarithmic equations is to rewrite the equation in exponential form or combine logarithms and then exponentiate. In either case, you must always check your solutions for extraneous values.

Strategy for Exponential Equations

1. Isolate the exponential expression on one side of the equation.
2. Take the logarithm (usually $\ln$ or $\log$) of both sides.
3. Use the power rule to bring down the exponent.
4. Solve for the variable.

Strategy for Logarithmic Equations

1. Use logarithm properties to combine into a single logarithm if needed.
2. Convert to exponential form: $\log_b(A) = C$ becomes $b^C = A$.
3. Solve for the variable.
4. Check that each solution makes the argument of every original logarithm positive (reject extraneous solutions).

4.5 Applications

Exponential and logarithmic functions model a wide variety of real-world phenomena. In this section, we focus on two of the most common: compound interest in finance and exponential decay in the sciences.

Compound Interest

When money earns interest on both the principal and the previously accumulated interest, the growth is exponential. The compound interest formula depends on how frequently the interest is compounded.

The continuous compounding formula is the limiting case as $n \to \infty$. It provides a slightly larger amount than any finite compounding frequency, but the difference is often small in practice.

Exponential Growth and Decay

Many natural processes follow an exponential model. If a quantity $Q$ changes at a rate proportional to its current value, then:

$$Q(t) = Q_0 \cdot e^{kt}$$

where $Q_0$ is the initial quantity at $t = 0$ and $k$ is the growth/decay constant. If $k > 0$, the quantity grows; if $k < 0$, it decays.

Half-Life

In radioactive decay, the half-life $h$ is the time it takes for a substance to reduce to half its initial amount. The relationship between $k$ and $h$ is:

$$Q_0 \cdot e^{kh} = \frac{Q_0}{2} \implies e^{kh} = \frac{1}{2} \implies k = \frac{-\ln(2)}{h}$$

4.6 Practice Problems

Test your understanding with the following problems. Each has a hidden solution you can reveal after attempting the problem on your own.