Lesson 5.10

Graphing Logarithms

The graph of is the reflection of across the line . It has a vertical asymptote at and passes through .

Introduction

Since the log is the inverse of the exponential, swapping x and y on gives , which is . Graphically, this means reflecting across the diagonal .

Past Knowledge

Graphing (5.2), transformations, inverse functions.

Today's Goal

Graph log functions, identify key features, apply transformations.

Future Success

Understanding domain restrictions helps avoid errors in log equations (Ch. 18).

Key Concepts

Inverse Relationship

Blue: · Red: · Dashed:

Key Features of

x-intercept: always, since

Vertical asymptote: (y-axis)

Domain: — only positive inputs

Range: — all real outputs

Passes through:

Worked Examples

Example 1: Plotting Points

Basic

Graph using a table.

x	1/9	1/3	1	3	9
y	−2	−1	0	1	2

Example 2: Transformed Log

Intermediate

Graph and identify the asymptote.

Shifts: right 3, up 1

Asymptote shifts from to . Domain:

Asymptote: , Domain:

Example 3: Reflection & Shift

Advanced

Graph and find the x-intercept.

Transformations: reflect over x-axis, up 2

The negative flips the curve upside-down. Asymptote stays at .

x-intercept: set y = 0

x-intercept:

Common Pitfalls

Vertical vs. Horizontal Asymptote

Exponential has a horizontal asymptote. Log has a vertical asymptote. They are mirror opposites across .

Domain Shifts

shifts the domain to . Forgetting the shift causes domain errors.

Real-Life Applications

Log scales compress huge ranges of data into manageable graphs. A log plot of earthquake magnitudes, sound levels, or star brightness makes patterns visible that would be invisible on a linear scale.