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Calculating interest rates, especially in the realm of continuously compounded interest, is a fascinating topic that revolves around a specific formula: \( A = Pe^{rt} \). In this formula:

• \( A \) represents the accumulated amount after time \( t \)
• \( P \) is the principal or initial amount
• \( r \) stands for the annual interest rate
• \( t \) is the time in years

Setting up the problem:
When knowing that your investment doubles, you plug \( A = 2P \) into the equation. This situation implies you are solving for when \( 2P = Pe^{rt} \) under the condition \( t = 15 \) years.
After simplifying, dividing both sides by \( P \) gets you to \( 2 = e^{15r} \). Next, you'll solve for \( r \) using the properties of natural logarithms.

Natural logarithms are a key mathematical concept used in solving equations related to exponential growth and decay, like those in continuously compounded interest.
Understanding the natural logarithm:
A logarithm is essentially the inverse of an exponent. A natural logarithm, symbolized as \( \ln \), specifically uses the base \( e \), which is an irrational constant approximately equal to 2.71828.
Applying natural logarithm in interest calculations:
To solve the equation \( 2 = e^{15r} \), take the natural logarithm on both sides to obtain \( \ln(2) = 15r \).
This simplifies our work: you use calculator tools to find that \( \ln(2) \approx 0.693 \). Therefore, you calculate the interest rate \( r \) using the expression \( r = \frac{\ln(2)}{15} \), leading us to an exact interest rate solution.

Doubling time is a practical concept that indicates how long it will take for an investment to double in value at a certain continuous interest rate. Understanding this concept helps in both short-term and long-term financial planning.
How doubling time is derived in continuous compounding:
It's all about rearranging the formula \( A = Pe^{rt} \) with \( A = 2P \) when accounting for doubling, and it simplifies under the continuous compounding formula scenario.
You reach the equation \( 2 = e^{15r} \), and by employing the natural logarithm, you derive \( \ln(2) = 15r \). This steps into finding the rate \( r = \frac{\ln(2)}{t} \), where \( t \) is the doubling time, given as 15 years in this example.
Calculating the time:
To reverse this thought, you can also determine the doubling time if the interest rate is known, using \( t = \frac{\ln(2)}{r} \), showing this vital relationship between time, rate, and doubling.