Lesson 8.2

Graphs of Logarithmic Functions

Since logarithms are inverses of exponentials, their graphs trade places: the domain becomes range, and horizontal asymptotes become vertical.

Introduction

The graph of is the reflection of across the line . This geometric relationship dictates every feature of the logarithmic graph.

Prerequisite Connection

Recall that has a horizontal asymptote at and domain .

Today's Increment

We swap and . The asymptote becomes vertical (), and the domain is restricted to .

Why This Matters

The visual of the vertical asymptote explains why diverges. The area under the curve grows just enough to be infinite.

Key Concepts

Parent Function

Vertical Asymptote: . The graph dives to as .

Intercept: Always .

Worked Examples

Example 1: Basic Graphing

Graph . Identify two key points.

Intercept Point

Let . Then . Point: (1, 0).

Base Point

Let

(the base). Then

. Point: (3, 1).

Example 2: Horizontal Shift

Find the domain and vertical asymptote of .

Find Argument Zero

Set .

Result

Asymptote:

. Domain:

Example 3: Reflection

Graph . Explain the transformation.

Transformation Logic

The negative sign is outside the function, so it reflects across the x-axis. Instead of growing to

, it dives to

Common Pitfalls

Touching the Asymptote

Students often draw the graph touching or crossing the y-axis. It never does! It gets infinitely close but never touches .

Real-World Application

Moore's Law

Moore's Law describes exponential growth in computing power. If we graph the "Year vs. Transistor Count" on a standard scale, it's a hockey stick. But if we graph "Year vs. Log(Transistors)", it becomes a straight line, making it much easier to predict future trends.