Exponential and logarithmic functions

Introduction to Exponential and Logarithmic Functions

Welcome to one of the most powerful chapters in IB Mathematics! Have you ever wondered how a virus spreads through a population, how money grows in a bank account, or how scientists measure the intensity of an earthquake? All these real-world scenarios rely on exponential and logarithmic functions.

In this chapter, we will explore how these two functions are "mirror images" of each other and learn how to master their graphs and equations. Don't worry if these seem intimidating at first—once you see the patterns, they become much easier to manage!

1. Exponential Functions: The Power of Growth

An exponential function is a function where the variable \(x\) is up in the exponent. The most common form you will see is \(f(x) = a^x\), where \(a\) is a positive number called the base.

Key Features of \(f(x) = a^x\)

• The \(y\)-intercept: Since any number (except zero) to the power of 0 is 1, the graph always passes through \((0, 1)\) unless it has been shifted.
• The Horizontal Asymptote: The graph gets closer and closer to the \(x\)-axis (\(y = 0\)) but never actually touches it. We call this a horizontal asymptote.
• Domain and Range: You can plug any value into \(x\), so the domain is all real numbers. However, the result \(a^x\) is always positive, so the range is \(f(x) > 0\).

Growth vs. Decay

• Exponential Growth: If the base \(a > 1\), the graph climbs upwards as it moves to the right. Think of this like a population doubling.
• Exponential Decay: If the base is between 0 and 1 (\(0 < a < 1\)), the graph falls as it moves to the right. Think of this like the value of a car decreasing over time.

The Special Number \(e\)

In the IB DP, you will frequently see the base \(e\). This is "Euler's number," and it is approximately \(2.718\). It works exactly like any other exponential base, but it is special because it occurs naturally in many growth processes. The function \(f(x) = e^x\) is often called the natural exponential function.

Quick Review: An exponential graph always stays above the \(x\)-axis and shoots up (or down) very quickly!

2. Logarithmic Functions: The Great Un-doer

A logarithm is simply the inverse of an exponential. If \(a^x = y\), then we say \(\log_a y = x\). Basically, a log asks the question: "What power do I need to raise the base to in order to get this number?"

Key Features of \(f(x) = \log_a x\)

Because logs are inverses of exponentials, their properties are "swapped":
• The \(x\)-intercept: The graph always passes through \((1, 0)\).
• The Vertical Asymptote: The graph gets closer and closer to the \(y\)-axis (\(x = 0\)) but never touches it. You cannot take the log of zero or a negative number!
• Domain and Range: The domain is \(x > 0\), and the range is all real numbers.

The Natural Logarithm

Just as \(e\) is the special base for exponentials, \(\ln x\) (the natural log) is the special log with base \(e\).
\(y = \ln x\) is the same as \(y = \log_e x\).

Did you know? Logarithms were invented before calculators to help sailors and astronomers do massive calculations by turning multiplication into simple addition!

3. The Relationship: Reflections in \(y = x\)

As you learned in the general functions chapter, inverse functions are reflections of each other over the line \(y = x\).
• If you take the graph of \(y = e^x\) and flip it over that diagonal line, you get \(y = \ln x\).
• This explains why the \(y\)-intercept \((0, 1)\) of the exponential becomes the \(x\)-intercept \((1, 0)\) of the log.

4. Solving Equations

The IB will ask you to solve equations involving these functions both analytically (by hand) and graphically (using your GDC).

Analytical Approach (Paper 1)

To solve an exponential equation by hand, you often need to "log both sides."
Example: Solve \(e^x = 10\)
1. Take the natural log of both sides: \(\ln(e^x) = \ln(10)\)
2. Use the property that \(\ln(e^x) = x\)
3. Final Answer: \(x = \ln(10)\)

Graphical Approach (Paper 2)

If the equation looks messy, like \(e^x = x + 5\), use your Graphic Display Calculator (GDC):
1. Plot \(y_1 = e^x\)
2. Plot \(y_2 = x + 5\)
3. Use the Intersect tool to find the \(x\)-coordinate(s) where the lines cross.

5. Real-Life Applications

Exponential functions are perfect for modeling. The general model is often written as:
\(f(t) = k a^t + c\) or \(f(t) = N_0 e^{rt}\)
• \(k\) or \(N_0\) is usually the initial value (what you start with at \(t = 0\)).
• \(r\) is the rate of growth or decay.
• \(c\) represents a shift, such as a horizontal asymptote representing the room temperature in a cooling problem.

Common Mistake to Avoid: When solving for time (\(t\)), students often forget that \(t\) must be positive in most contexts. Always check if your answer makes sense in the real world!

Summary: Key Takeaways

Exponential Functions (\(y = a^x\)):
• Horizontal asymptote at \(y = 0\).
• Growth if \(a > 1\), decay if \(0 < a < 1\).
• Passes through \((0, 1)\).

Logarithmic Functions (\(y = \log_a x\)):
• Vertical asymptote at \(x = 0\).
• Domain is only positive numbers (\(x > 0\)).
• Passes through \((1, 0)\).

Working with Base \(e\):
• \(e\) and \(\ln\) are "opposites" and cancel each other out. This is your best tool for solving equations!