Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![Below is the transcription of the provided image and a detailed explanation relevant to solving the equation:
**Transcription:**
5) \( 5e^{3x+4} = 15 \)
**Explanation for Educational Website:**
Solve the equation:
Given the equation:
\[ 5e^{3x+4} = 15 \]
To solve for \( x \), follow these steps:
1. **Isolate the exponential term:**
Divide both sides of the equation by 5:
\[ e^{3x+4} = \frac{15}{5} \]
Simplify the right side:
\[ e^{3x+4} = 3 \]
2. **Apply the natural logarithm (ln) to both sides:**
Because the natural logarithm is the inverse of the exponential function, applying it will help to isolate the exponent.
\[ \ln(e^{3x+4}) = \ln(3) \]
3. **Simplify using the property of logarithms:**
According to the property of logarithms \( \ln(e^y) = y \):
\[ 3x + 4 = \ln(3) \]
4. **Solve for \( x \):**
Isolate \( x \) by first subtracting 4 from both sides:
\[ 3x = \ln(3) - 4 \]
Then, divide by 3:
\[ x = \frac{\ln(3) - 4}{3} \]
Therefore, the solution for \( x \) is:
\[ x = \frac{\ln(3) - 4}{3} \]
**Graphical Interpretation:**
Typically, a graph of \( y = e^{3x+4} \) would show an exponential growth curve, and a horizontal line for \( y = 3 \). The intersection point of these two graphs would represent the solution to the equation. Solving the equation algebraically, as shown, will provide the exact x-coordinate of this intersection point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F123ebf30-385c-4851-84fd-382b6e3a9385%2F41bf69be-9b9a-4a3c-9b9c-7297b32036ce%2F618zcxp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription of the provided image and a detailed explanation relevant to solving the equation:
**Transcription:**
5) \( 5e^{3x+4} = 15 \)
**Explanation for Educational Website:**
Solve the equation:
Given the equation:
\[ 5e^{3x+4} = 15 \]
To solve for \( x \), follow these steps:
1. **Isolate the exponential term:**
Divide both sides of the equation by 5:
\[ e^{3x+4} = \frac{15}{5} \]
Simplify the right side:
\[ e^{3x+4} = 3 \]
2. **Apply the natural logarithm (ln) to both sides:**
Because the natural logarithm is the inverse of the exponential function, applying it will help to isolate the exponent.
\[ \ln(e^{3x+4}) = \ln(3) \]
3. **Simplify using the property of logarithms:**
According to the property of logarithms \( \ln(e^y) = y \):
\[ 3x + 4 = \ln(3) \]
4. **Solve for \( x \):**
Isolate \( x \) by first subtracting 4 from both sides:
\[ 3x = \ln(3) - 4 \]
Then, divide by 3:
\[ x = \frac{\ln(3) - 4}{3} \]
Therefore, the solution for \( x \) is:
\[ x = \frac{\ln(3) - 4}{3} \]
**Graphical Interpretation:**
Typically, a graph of \( y = e^{3x+4} \) would show an exponential growth curve, and a horizontal line for \( y = 3 \). The intersection point of these two graphs would represent the solution to the equation. Solving the equation algebraically, as shown, will provide the exact x-coordinate of this intersection point.
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