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Graphing Logarithmic Functions

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Graphing Logarithmic Functions

Study Guide

Key Definition

The function $y = \log_b x$ is the inverse of the exponential function $y = b^x$.

Important Notes

The graph of the inverse function is the reflection about the line $y = x$.
The domain of $y = \log_b x$ is all positive real numbers.
Vertical shift: If $k > 0$, shift up; if $k < 0$, shift down.
Horizontal shift: For $y = \log_b(x - h)$, h > 0 shifts right; for $y = \log_b(x + h)$, h > 0 shifts left.
Vertical asymptote: The graph has a vertical asymptote at x = 0 for $y = \log_b(x)$ and at x = h for $y = \log_b(x - h)$.
Range: All real numbers.
Base restrictions: b > 0 and b ≠ 1.
X-intercept: For $y = \log_b x$, the graph crosses the x-axis at (1, 0).
Behavior: As x → 0⁺, y → -∞; as x → ∞, y → ∞.
Reflections and stretching: Reflections across the y-axis and vertical stretches/compressions occur when the function includes factors like -$\log_b x$ or a·$\log_b x$.
Assume base 10 if the base is not specified.

Mathematical Notation

$\log_b x$ represents the logarithm of $x$ with base $b$.

$+$ represents addition

$-$ represents subtraction

$( )$ denote the argument of the function or grouping

$x = h$ denotes a vertical asymptote

Use '+' and '-' for shifts, parentheses to denote function arguments, and 'x = h' to indicate vertical asymptotes.

Why It Works

The logarithm $y = \log_b(x)$ is the inverse of y = b^x. Graphically, each point (x, y) on the exponential graph corresponds to (y, x) on the logarithm graph, causing a reflection across the line $y = x$.

Remember

The graph of $y = \log_b x$ is the reflection of y = b^x across the line $y = x$.

Quick Reference

Vertical Shift:$y = \log_b(x) + k$ shifts graph up by k units if k > 0, down if k < 0.

Horizontal Shift:$y = \log_b(x - h)$ shifts right by h units; $y = \log_b(x + h)$ shifts left by h units.

Understanding Graphing Logarithmic Functions

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Video explanation of this concept

Beginner

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Beginner Explanation

The basic logarithmic function $y = \log_b(x)$ passes through (1, 0), has domain x > 0, range all real numbers, and a vertical asymptote at x = 0. As x increases, y increases slowly.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the domain of the function $y = \log_3 x$?

Real-World Problem

Question Exercise

Intermediate

Scientist Scenario

A scientist uses $y = \log_{10} x$ to measure earthquake magnitudes. Explain the graph's shifts when $y = \log_{10}(x - 2) + 3$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explain how $y = \log_2(x + 1) - 3$ is a transformation of $y = \log_2 x$.

Challenge Quiz

Single Choice Quiz

Advanced

Which transformation is applied to $y = \log_5 x$ to get $y = \log_5(x - 4) + 2$?

Recap

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