AP Precalculus: Logarithmic Functions

Master graphing, transformations, domain, range, and the inverse relationship with exponentials

📊 Graphing 🔄 Transformations 📏 Asymptotes ↔️ Inverses

📚 Understanding Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. While exponential functions answer "what is \(b\) raised to power \(x\)?", logarithmic functions answer "what power gives us this value?" This deep connection means their graphs are reflections of each other across the line \(y = x\).

1 Logarithmic Function General Form

A logarithmic function with transformations has the general form shown below. Each parameter controls a specific aspect of the graph.

General Transformed Form \[f(x) = a \cdot \log_b(x - h) + k\]

Understanding Each Parameter

\(a\) — Vertical Stretch/Reflect

\(|a| > 1\): stretch vertically
\(0 < |a| < 1\): compress
\(a < 0\): reflect over x-axis

\(b\) — Base

Must be \(b > 0\) and \(b \neq 1\)
\(b > 1\): increasing function
\(0 < b < 1\): decreasing function

\(h\) — Horizontal Shift

\(h > 0\): shift right
\(h < 0\): shift left
Moves vertical asymptote

\(k\) — Vertical Shift

\(k > 0\): shift up
\(k < 0\): shift down
Does not affect asymptote

📌 Example

Identify the parameters of: \(f(x) = -2\log_3(x + 4) - 1\)

\(a = -2\) (reflected over x-axis, vertically stretched by 2)

\(b = 3\) (base 3 logarithm)

\(h = -4\) (shifted left 4 units)

\(k = -1\) (shifted down 1 unit)

2 Domain, Range, and Asymptote

Unlike exponential functions (which have horizontal asymptotes), logarithmic functions have vertical asymptotes. The domain is restricted because you can only take the log of positive numbers.

📐 Domain

The argument must be positive:
\(x - h > 0\)
Domain: \(x > h\) or \((h, \infty)\)

📏 Range

Logarithms can output any real number:
Range: \((-\infty, \infty)\) or all real numbers
(No restrictions on y-values)

📍 Vertical Asymptote

The graph approaches but never crosses:
Asymptote: \(x = h\)
(Shifts with horizontal transformation)

📌 Examples

\(f(x) = \log_2 x\): Domain: \(x > 0\), Range: all reals, VA: \(x = 0\)

\(f(x) = \log(x - 3)\): Domain: \(x > 3\), Range: all reals, VA: \(x = 3\)

\(f(x) = \ln(x + 5) + 2\): Domain: \(x > -5\), Range: all reals, VA: \(x = -5\)

⚠️ Finding Domain

Always set the argument of the logarithm \(> 0\) and solve for \(x\). Remember: there is no log of zero or negative numbers!

3 Parent Logarithmic Functions

Before applying transformations, understand the basic (parent) logarithmic functions. All log functions share the same general shape but differ in their steepness.

\(f(x) = \log_b x\) (Base \(b > 1\))

• Passes through \((1, 0)\)
• Increasing function
• Passes through \((b, 1)\)
• VA at \(x = 0\)

\(f(x) = \log x\) (Common Log)

• Base 10
• Passes through \((1, 0)\) and \((10, 1)\)
• Used in science, engineering
• Calculator: LOG button

\(f(x) = \ln x\) (Natural Log)

• Base \(e ≈ 2.718\)
• Passes through \((1, 0)\) and \((e, 1)\)
• Used in calculus
• Calculator: LN button

Key Points on Any Parent Log Function \(y = \log_b x\)

\((1, 0)\) — always passes through this point (since \(\log_b 1 = 0\))
\((b, 1)\) — passes through this point (since \(\log_b b = 1\))
\((b^2, 2)\) — another easy point (since \(\log_b b^2 = 2\))
\((\frac{1}{b}, -1)\) — on the curve (since \(\log_b \frac{1}{b} = -1\))

4 Graphing Logarithmic Functions

To graph a transformed logarithmic function, start with the parent function and apply transformations in the correct order.

Steps to Graph \(f(x) = a \cdot \log_b(x - h) + k\)

Identify the vertical asymptote: \(x = h\) (draw a dashed vertical line)
Find the "anchor point": The point \((h + 1, k)\) replaces \((1, 0)\) from the parent
Find a second point: Use \((h + b, a + k)\) which corresponds to the old \((b, 1)\)
Apply reflection if \(a < 0\): Points flip across the horizontal line \(y = k\)
Sketch the curve: Approaches asymptote on one side, continues rising/falling on the other

📌 Example: Graph \(f(x) = \log_2(x - 3) + 1\)

Step 1: Vertical asymptote at \(x = 3\)

Step 2: Anchor point: \((3 + 1, 0 + 1) = (4, 1)\)

Step 3: Second point: \((3 + 2, 1 + 1) = (5, 2)\)

Step 4: No reflection (\(a = 1 > 0\))

Domain: \(x > 3\), Range: all reals

5 Transformations Summary

Here's a complete reference for how each transformation affects the logarithmic function graph.

Transformation	Equation	Effect on Graph
Shift Right \(h\)	\(\log_b(x - h)\)	Graph moves right; VA moves to \(x = h\)
Shift Left \(h\)	\(\log_b(x + h)\)	Graph moves left; VA moves to \(x = -h\)
Shift Up \(k\)	\(\log_b x + k\)	Graph moves up; VA unchanged
Shift Down \(k\)	\(\log_b x - k\)	Graph moves down; VA unchanged
Vertical Stretch	\(a \cdot \log_b x\), \(\|a\| > 1\)	Graph stretched away from x-axis
Vertical Compression	\(a \cdot \log_b x\), \(0 < \|a\| < 1\)	Graph compressed toward x-axis
Reflect over x-axis	\(-\log_b x\)	Graph flips upside down
Reflect over y-axis	\(\log_b(-x)\)	Graph reflects left; Domain: \(x < 0\)

💡 Order of Transformations

Apply transformations inside-to-outside: (1) Horizontal shift \(h\), (2) Horizontal dilation, (3) Vertical dilation/reflection \(a\), (4) Vertical shift \(k\).

6 Inverse Relationship with Exponentials

Logarithmic and exponential functions are inverses of each other. Their graphs are reflections across the line \(y = x\), and they "undo" each other algebraically.

Exponential Function

\(y = b^x\)

⟺

Logarithmic Function

\(y = \log_b x\)

Comparing Properties

Property	Exponential \(y = b^x\)	Logarithmic \(y = \log_b x\)
Domain	All real numbers	\(x > 0\)
Range	\(y > 0\)	All real numbers
Asymptote	Horizontal: \(y = 0\)	Vertical: \(x = 0\)
Key Point	\((0, 1)\)	\((1, 0)\)
Intercept	Y-intercept at 1	X-intercept at 1

📌 Graphs Are Reflections

If you plot \(y = 2^x\) and \(y = \log_2 x\) on the same axes, they are mirror images across the line \(y = x\).

Notice: \((0, 1)\) on \(2^x\) corresponds to \((1, 0)\) on \(\log_2 x\) — the coordinates swap!

7 Finding the Equation from a Graph

Given a graph of a logarithmic function, you can determine its equation by identifying key features.

Steps to Find the Equation

Find the vertical asymptote — this gives you \(h\) (from \(x = h\))
Find a point on the graph — preferably one unit to the right of the asymptote
Identify the y-value at \(x = h + 1\) — this gives you \(k\) (since \(\log_b 1 = 0\))
Check for reflection — is the function increasing (normal) or decreasing (reflected)?
Use another point to find \(a\) and \(b\) if needed

📌 Example

Given: A log graph with VA at \(x = 2\), passes through \((3, 4)\) and \((5, 5)\)

From VA: \(h = 2\)

At \(x = 3 = h + 1\): \(y = 4 = k\) (so \(k = 4\))

Partial equation: \(f(x) = a \cdot \log_b(x - 2) + 4\)

Using \((5, 5)\): \(5 = a \cdot \log_b(3) + 4\) → \(a \cdot \log_b 3 = 1\)

If base 3: \(a \cdot 1 = 1\) → \(a = 1\)

Equation: \(f(x) = \log_3(x - 2) + 4\)

8 Matching Functions and Graphs

On the AP exam, you may need to match logarithmic equations to their graphs. Use these quick checks:

Quick Identification Guide

Vertical asymptote location: Tells you the value of \(h\)
Direction of curve: Increasing (normal) vs. decreasing (reflected by \(a < 0\) or \(0 < b < 1\))
Steepness: \(|a| > 1\) = steeper; \(|a| < 1\)=flatter
Position relative to x-axis: Helps identify \(k\) (vertical shift)
X-intercept: Located at \(x = h + b^{-k/a}\)

💡 Quick Test

To find the x-intercept, set \(y = 0\) and solve: \(0 = a\log_b(x-h) + k\) gives \(x = h + b^{-k/a}\)

📋 Quick Reference: Key Formulas

General Form

\(f(x) = a \cdot \log_b(x - h) + k\)

Domain

\(x > h\) (argument > 0)

Range

All real numbers \((-\infty, \infty)\)

Vertical Asymptote

\(x = h\)

Key Point

\((h + 1, k)\) — anchor point

Inverse

\(y = \log_b x \Leftrightarrow y = b^x\)