2. The points (1, 0) and (2, 1) are on the graph of f (x) = log2 x. a) Find exact coordinates of two other points that are on the graph of f(x) = log, x. b) Using transformations, sketch a graph of g(x) = 3 log2(x + 1). Draw and label your asymptote. USE &
2. The points (1, 0) and (2, 1) are on the graph of f (x) = log2 x. a) Find exact coordinates of two other points that are on the graph of f(x) = log, x. b) Using transformations, sketch a graph of g(x) = 3 log2(x + 1). Draw and label your asymptote. USE &
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Mathematics - Logarithmic Functions
#### Problem 2: Analysis and Graphing of Logarithmic Functions
**Given:**
The points \((1, 0)\) and \((2, 1)\) are on the graph of \( f(x) = \log_2 x \).
**Part a)**
**Objective:** Find the exact coordinates of two other points that are on the graph of \( f(x) = \log_2 x \).
**Solution:**
To find additional points on the graph of \( f(x) = \log_2 x \), we can select \( x \) values for which the logarithm base 2 is easy to compute. For example:
- When \( x = 4 \):
\[ f(4) = \log_2 4 = 2 \]
Hence, the point \((4, 2)\) is on the graph.
- When \( x = \frac{1}{2} \):
\[ f\left( \frac{1}{2} \right) = \log_2 \left( \frac{1}{2} \right) = -1 \]
Hence, the point \(\left( \frac{1}{2}, -1 \right)\) is on the graph.
So, the two additional points are \( (4, 2) \) and \( \left( \frac{1}{2}, -1 \right) \).
**Part b)**
**Objective:** Using transformations, sketch a graph of \( g(x) = 3 \log_2 (x + 1) \). Draw and label your asymptote.
**Steps for Transformation:**
The function \( g(x) = 3 \log_2 (x + 1) \) can be derived from \( f(x) = \log_2 x \) by the following transformations:
1. **Horizontal Shift:** The term \( (x + 1) \) indicates a shift left by 1 unit.
2. **Vertical Stretch:** The coefficient 3 before the logarithm function indicates a vertical stretch by a factor of 3.
**Asymptote:**
The original function \( f(x) = \log_2 x \) has a vertical asymptote at \( x = 0 \). After the horizontal shift by 1 unit to the left, the asympt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7aed86dd-d5c3-48c6-a690-9abed80f4c22%2Fddffc323-764b-492d-ba0d-d3c269629e38%2Fpzcs3ok.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematics - Logarithmic Functions
#### Problem 2: Analysis and Graphing of Logarithmic Functions
**Given:**
The points \((1, 0)\) and \((2, 1)\) are on the graph of \( f(x) = \log_2 x \).
**Part a)**
**Objective:** Find the exact coordinates of two other points that are on the graph of \( f(x) = \log_2 x \).
**Solution:**
To find additional points on the graph of \( f(x) = \log_2 x \), we can select \( x \) values for which the logarithm base 2 is easy to compute. For example:
- When \( x = 4 \):
\[ f(4) = \log_2 4 = 2 \]
Hence, the point \((4, 2)\) is on the graph.
- When \( x = \frac{1}{2} \):
\[ f\left( \frac{1}{2} \right) = \log_2 \left( \frac{1}{2} \right) = -1 \]
Hence, the point \(\left( \frac{1}{2}, -1 \right)\) is on the graph.
So, the two additional points are \( (4, 2) \) and \( \left( \frac{1}{2}, -1 \right) \).
**Part b)**
**Objective:** Using transformations, sketch a graph of \( g(x) = 3 \log_2 (x + 1) \). Draw and label your asymptote.
**Steps for Transformation:**
The function \( g(x) = 3 \log_2 (x + 1) \) can be derived from \( f(x) = \log_2 x \) by the following transformations:
1. **Horizontal Shift:** The term \( (x + 1) \) indicates a shift left by 1 unit.
2. **Vertical Stretch:** The coefficient 3 before the logarithm function indicates a vertical stretch by a factor of 3.
**Asymptote:**
The original function \( f(x) = \log_2 x \) has a vertical asymptote at \( x = 0 \). After the horizontal shift by 1 unit to the left, the asympt
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