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Exponential growth is a process where a quantity increases over time at a rate proportional to its current value. This means the larger the quantity, the faster it grows. The function for exponential growth can be expressed as:

\( A = P e^{rt} \)

• \( A \) is the amount of money accumulated after time \( t \)
• \( P \) is the principal amount (initial investment)
• \( r \) is the annual interest rate
• \( t \) is the time in years
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

This formula is particularly relevant to problems involving continuously compounded interest.

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \). It's a fundamental concept in mathematics, especially in calculus and exponential growth equations. The natural logarithm of a number \( x \) gives the power to which \( e \) must be raised to obtain \( x \). Mathematically, if \( y = \ln(x) \), then \( e^y = x \).

For example, to solve the equation \( A = P e^{rt} \) for \( r \), we can take the natural logarithm of both sides to make it manageable:

\( \ln(A) = \ln(P) + rt \)

• Using the logarithmic rules makes it easier to isolate the variable \( r \).
• Once \( r \) is isolated, we can solve for it using algebraic methods.
• This step is crucial in problems involving continuous compounding.

In this context, solving equations involves isolating the variable of interest. Here are the steps to solve the exponential growth problem:

• Identify the given variables: We start with the equation \( A = P e^{rt} \).
• Substitute known values: Since the amount doubles, \( A = 2P \). For a principal amount of \$1000 doubling in 3 years, we have \( 2P = P e^{3r} \).
• Use natural logarithms: Divide both sides by \( P \), giving \( 2 = e^{3r} \). Taking the natural logarithm of both sides, we get \( \ln(2) = 3r \).
• Solve for \( r \): Finally, isolate \( r \) by dividing both sides by 3: \( r = \frac{\ln(2)}{3} = 0.23105 \approx 23.11\% \)

This process helps in understanding how to handle problems involving continuous compounding and exponential relationships.