Chapter6: Exponential And Logarithmic Functions
Section6.1: Exponential Functions
Problem 68SE: An investment account with an annual interest rateof 7 was opened with an initial deposit of 4,000...
Related questions
Question
Solve the equation?
![**Transcription for Educational Use**
---
**Problem 9**: Solve the equation for the variable \( r \).
\[ A = Pe^{rt} \]
---
**Explanation**:
The equation given is commonly used in situations involving continuous compound interest or exponential growth and decay. Here:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
To solve for \( r \), one would typically follow these steps:
1. Divide both sides by \( P \) to isolate the exponential expression:
\[ \frac{A}{P} = e^{rt} \]
2. Take the natural logarithm (ln) of both sides to eliminate the base \( e \):
\[ \ln\left(\frac{A}{P}\right) = rt \]
3. Solve for \( r \) by dividing both sides by \( t \):
\[ r = \frac{\ln\left(\frac{A}{P}\right)}{t} \]
This equation now expresses \( r \) in terms of the other given variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e9ec804-819a-427a-935b-9e49b0134ee5%2F2368e28f-f7dc-44c8-9f87-c751f39726ac%2Fgc80tg6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Use**
---
**Problem 9**: Solve the equation for the variable \( r \).
\[ A = Pe^{rt} \]
---
**Explanation**:
The equation given is commonly used in situations involving continuous compound interest or exponential growth and decay. Here:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
To solve for \( r \), one would typically follow these steps:
1. Divide both sides by \( P \) to isolate the exponential expression:
\[ \frac{A}{P} = e^{rt} \]
2. Take the natural logarithm (ln) of both sides to eliminate the base \( e \):
\[ \ln\left(\frac{A}{P}\right) = rt \]
3. Solve for \( r \) by dividing both sides by \( t \):
\[ r = \frac{\ln\left(\frac{A}{P}\right)}{t} \]
This equation now expresses \( r \) in terms of the other given variables.
Expert Solution
Step 1
Given,
Step by step
Solved in 2 steps
Recommended textbooks for you