a) Use properties of logarithms to simplify the function f(x) = In(e5*) + elnx – 2x – 3 In 1+4 ln e. State the domain of f(x). b) Graph the function f(x).
a) Use properties of logarithms to simplify the function f(x) = In(e5*) + elnx – 2x – 3 In 1+4 ln e. State the domain of f(x). b) Graph the function f(x).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem
a) Use properties of logarithms to simplify the function
\[ f(x) = \ln (e^{5x}) + e^{\ln x} - 2x - 3 \ln 1 + 4 \ln e. \]
State the domain of \( f(x) \).
b) Graph the function \( f(x) \).
### Simplification and Domain
#### Simplification:
1. \( \ln (e^{5x}) = 5x \) because \( \ln (e^y) = y \).
2. \( e^{\ln x} = x \) because \( e^{\ln y} = y \).
3. \( \ln 1 = 0 \) because any number to the power of 0 is 1.
4. \( \ln e = 1 \) because the natural log of \( e \) is 1.
Now, substituting these into \( f(x) \):
\[ f(x) = 5x + x - 2x - 3(0) + 4(1) \]
\[ f(x) = 5x + x - 2x + 4 \]
\[ f(x) = 4x + 4 \]
#### Domain:
The domain of \( f(x) = 4x + 4 \) is all real numbers, since there are no restrictions from logarithms when simplified.
### Graph Explanation
There is a blank graph with labeled axes likely designed for students to sketch the linear function \( f(x) \).
To graph \( f(x) = 4x + 4 \):
- The y-intercept (where \( x = 0 \)) is \( 4 \).
- The slope is \( 4 \), meaning the graph rises 4 units for every 1 unit it runs to the right.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f6e2229-f08e-467f-a4fe-3e692a6a0806%2Ff2d82ac6-fa12-48a4-9c6f-a4c33c0b9ac6%2Frsgvwea_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem
a) Use properties of logarithms to simplify the function
\[ f(x) = \ln (e^{5x}) + e^{\ln x} - 2x - 3 \ln 1 + 4 \ln e. \]
State the domain of \( f(x) \).
b) Graph the function \( f(x) \).
### Simplification and Domain
#### Simplification:
1. \( \ln (e^{5x}) = 5x \) because \( \ln (e^y) = y \).
2. \( e^{\ln x} = x \) because \( e^{\ln y} = y \).
3. \( \ln 1 = 0 \) because any number to the power of 0 is 1.
4. \( \ln e = 1 \) because the natural log of \( e \) is 1.
Now, substituting these into \( f(x) \):
\[ f(x) = 5x + x - 2x - 3(0) + 4(1) \]
\[ f(x) = 5x + x - 2x + 4 \]
\[ f(x) = 4x + 4 \]
#### Domain:
The domain of \( f(x) = 4x + 4 \) is all real numbers, since there are no restrictions from logarithms when simplified.
### Graph Explanation
There is a blank graph with labeled axes likely designed for students to sketch the linear function \( f(x) \).
To graph \( f(x) = 4x + 4 \):
- The y-intercept (where \( x = 0 \)) is \( 4 \).
- The slope is \( 4 \), meaning the graph rises 4 units for every 1 unit it runs to the right.
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