5e3x+4 = 15
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Solve each equation for the given variable and express your answer to the nearest hundredth.
![### Solving Exponential Equations
#### Example Problem 5:
Solve the following exponential equation for \( x \):
\[ 5e^{3x + 4} = 15 \]
### Steps to Solve:
1. **Isolate the exponential term:**
Divide both sides of the equation by 5:
\[
e^{3x + 4} = 3
\]
2. **Apply the natural logarithm to both sides:**
To eliminate the exponential term, take the natural logarithm (ln) of both sides:
\[
\ln(e^{3x + 4}) = \ln(3)
\]
3. **Simplify using the properties of logarithms:**
Use the property \(\ln(e^y) = y\):
\[
3x + 4 = \ln(3)
\]
4. **Solve for \( x \):**
- Subtract 4 from both sides:
\[
3x = \ln(3) - 4
\]
- Divide by 3:
\[
x = \frac{\ln(3) - 4}{3}
\]
### Solution:
The solution to the equation \( 5e^{3x + 4} = 15 \) is:
\[
x = \frac{\ln(3) - 4}{3}
\]
### Explanation:
This problem demonstrates the process of solving exponential equations by isolating the exponential term, applying natural logarithms, and then performing algebraic operations to solve for the unknown variable. This technique is useful in various mathematical and real-world applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32185d28-13fb-46f8-83ec-ad4b91fe80d7%2F69241e25-7325-424a-b0c8-7dbab2dc7dbf%2Fbpz80jp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations
#### Example Problem 5:
Solve the following exponential equation for \( x \):
\[ 5e^{3x + 4} = 15 \]
### Steps to Solve:
1. **Isolate the exponential term:**
Divide both sides of the equation by 5:
\[
e^{3x + 4} = 3
\]
2. **Apply the natural logarithm to both sides:**
To eliminate the exponential term, take the natural logarithm (ln) of both sides:
\[
\ln(e^{3x + 4}) = \ln(3)
\]
3. **Simplify using the properties of logarithms:**
Use the property \(\ln(e^y) = y\):
\[
3x + 4 = \ln(3)
\]
4. **Solve for \( x \):**
- Subtract 4 from both sides:
\[
3x = \ln(3) - 4
\]
- Divide by 3:
\[
x = \frac{\ln(3) - 4}{3}
\]
### Solution:
The solution to the equation \( 5e^{3x + 4} = 15 \) is:
\[
x = \frac{\ln(3) - 4}{3}
\]
### Explanation:
This problem demonstrates the process of solving exponential equations by isolating the exponential term, applying natural logarithms, and then performing algebraic operations to solve for the unknown variable. This technique is useful in various mathematical and real-world applications.
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