Chapter 6: Exponential & Logarithmic Functions

Exponential and logarithmic functions are among the most important tools in mathematics. They model phenomena that grow or shrink at rates proportional to their current size: populations doubling every generation, radioactive substances losing half their mass at regular intervals, investments compounding over time. In this chapter we build a thorough understanding of these functions, learn how to manipulate and solve equations involving them, and apply them to real-world problems in finance, biology, chemistry, and physics.

6.1 Exponential Functions

Up to this point in Algebra 2, the functions we have studied—linear, quadratic, polynomial—have the variable $x$ appearing in the base with a constant exponent. Exponential functions reverse this relationship: the variable appears in the exponent.

The restriction $b > 0$ ensures the function is defined for all real numbers $x$. We also exclude $b = 1$ because $1^x = 1$ for all $x$, which produces the constant function $f(x) = a$ rather than an exponential.

Growth vs. Decay

The behavior of an exponential function depends entirely on whether the base $b$ is greater than or less than 1:

• Exponential Growth ($b > 1$): As $x$ increases, $b^x$ increases without bound. The function rises steeply to the right. Common examples include population growth and compound interest.
• Exponential Decay ($0 < b < 1$): As $x$ increases, $b^x$ approaches 0. The function decreases toward the $x$-axis but never reaches it. Common examples include radioactive decay and depreciation.

Graphing and Transformations

We can apply the standard toolkit of transformations to exponential functions. Given a parent function $f(x) = b^x$, the general transformed form is:

$$g(x) = a \cdot b^{x - h} + k$$

Each parameter controls a specific transformation:

• $a$: Vertical stretch (if $|a| > 1$) or compression (if $0 < |a| < 1$). If $a < 0$, the graph reflects over the $x$-axis.
• $h$: Horizontal shift. The graph moves $h$ units to the right.
• $k$: Vertical shift. The horizontal asymptote moves from $y = 0$ to $y = k$.

The Natural Base $e$

Among all possible bases for exponential functions, one stands out as the most mathematically natural. The number $e$, known as Euler's number, is defined as

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

The function $f(x) = e^x$ has a remarkable property: at every point, its rate of change equals its current value. This makes $e^x$ the natural choice for modeling continuous growth and decay processes. In calculus, you will learn that the derivative of $e^x$ is simply $e^x$ itself—no other base has this property.

6.2 Logarithmic Functions

Because exponential functions are one-to-one (they pass the horizontal line test), each exponential function has an inverse. That inverse is the logarithmic function.

This definition establishes a direct correspondence between exponential and logarithmic forms. Being able to convert fluently between these two forms is essential for solving equations.

Common Log and Natural Log

Two logarithmic bases appear so frequently that they have special notation:

• Common logarithm: $\log x = \log_{10} x$. Used in pH calculations, decibel scales, and the Richter scale.
• Natural logarithm: $\ln x = \log_e x$. Used throughout calculus, physics, and continuous growth models.

Your calculator has dedicated buttons for both: LOG for $\log_{10}$ and LN for $\ln$.

Since $y = \log_b x$ is the inverse of $y = b^x$, their graphs are reflections of each other across the line $y = x$. This geometric relationship provides immediate insight: any horizontal asymptote of the exponential becomes a vertical asymptote of the logarithm, and vice versa.

6.3 Properties of Logarithms

Because logarithms are exponents, the familiar exponent rules translate directly into logarithm rules. These properties allow us to expand, condense, and simplify logarithmic expressions—skills that are essential for solving equations in the next two sections.

These properties follow directly from the corresponding exponent laws. For instance, the product rule mirrors the fact that $b^m \cdot b^n = b^{m+n}$. If $\log_b M = m$ and $\log_b N = n$, then $MN = b^m \cdot b^n = b^{m+n}$, so $\log_b(MN) = m + n = \log_b M + \log_b N$.

Change of Base Formula

Since most calculators only compute $\log$ (base 10) and $\ln$ (base $e$), we need a way to evaluate logarithms with any base. The change of base formula provides this.

6.4 Solving Exponential Equations

An exponential equation is one in which the variable appears in an exponent. There are two main strategies for solving these equations, depending on whether the two sides can be written with the same base.

Strategy 1: Same Base

If both sides of an equation can be expressed as powers of the same base, then we can use the one-to-one property:

Strategy 2: Using Logarithms

When the two sides cannot be rewritten with a common base, we take the logarithm of both sides. You can use either $\log$ or $\ln$; both give the same answer.

6.5 Solving Logarithmic Equations

A logarithmic equation contains a logarithm of an expression involving the variable. The key strategy is to convert from logarithmic form to exponential form, or to use properties of logarithms to combine multiple logarithms into one.

Strategy 1: Convert to Exponential Form

Strategy 2: Use Properties to Combine

When an equation has multiple logarithmic terms, use the product, quotient, or power rules to combine them into a single logarithm, then convert to exponential form.

6.6 Applications

Exponential and logarithmic functions are not abstract curiosities—they model some of the most important quantitative relationships in the world. In this section we explore four major application areas.

Compound Interest

The more frequently interest is compounded, the more the investment grows. As $n \to \infty$, we arrive at continuous compounding:

Population Growth

Many biological populations grow exponentially when resources are plentiful. The standard model is:

$$P(t) = P_0 \cdot e^{kt}$$

where $P_0$ is the initial population and $k$ is the growth rate constant. If $k > 0$, the population is growing; if $k < 0$, it is declining.

Radioactive Decay and Half-Life

Radioactive substances decay at a rate proportional to the amount present, following the model:

$$A(t) = A_0 \cdot e^{-kt}$$

The half-life $t_{1/2}$ is the time required for half the substance to decay. Setting $A(t_{1/2}) = \frac{1}{2}A_0$:

$$\frac{1}{2} = e^{-kt_{1/2}} \quad \Longrightarrow \quad t_{1/2} = \frac{\ln 2}{k}$$

The pH Scale

The pH of a solution is a logarithmic measure of its hydrogen ion concentration $[\text{H}^+]$, measured in moles per liter:

$$\text{pH} = -\log[\text{H}^+]$$

Because the scale is logarithmic, each whole number change in pH represents a tenfold change in hydrogen ion concentration. A solution with pH 3 is ten times more acidic than one with pH 4.

6.7 Practice Problems

Work through the following problems to solidify your understanding of exponential and logarithmic functions. Each problem has a detailed solution that you can reveal after attempting it yourself.