Problem 369 - Precalculus

Question

Recall the formula for continually compounding interest, \(y=A e^{k t} .\) Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

Step-by-Step Solution

Answer

The time \( t \) is given by \( t = \frac{\ln \left( \frac{y}{A} \right)}{k} \).

Step 1: Understanding the Formula

The given formula for continually compounded interest is \( y = A e^{kt} \). Here, \( y \) is the final amount, \( A \) is the initial amount, \( k \) is the rate, and \( t \) is the time we need to solve for.

Step 2: Isolating the Exponential Term

First, we need to isolate the exponential term on one side of the equation. We can do this by dividing both sides of the equation by \( A \): \( \frac{y}{A} = e^{kt} \).

Step 3: Applying the Logarithm to Both Sides

To solve for \( t \), take the natural logarithm (ln) of both sides of the equation: \( \ln \left( \frac{y}{A} \right) = \ln(e^{kt}) \).

Step 4: Using Logarithm Properties

Use the property of logarithms that \( \ln(e^x) = x \) to simplify the right side: \( \ln \left( \frac{y}{A} \right) = kt \).

Step 5: Solving for Time \( t \)

Finally, solve for \( t \) by dividing both sides by \( k \): \[ t = \frac{\ln \left( \frac{y}{A} \right)}{k} \].

Key Concepts

Continually Compounding Interest

Continually compounding interest is a concept from finance and mathematics that explains how an amount grows exponentially over time when it's compounded continuously. This is different from regular compounding, where interest is added at standard intervals like annually or quarterly. In the formula \( y = A e^{kt} \), \( A \) represents the initial investment or principal, \( e \) is the constant approximately equal to 2.71828, \( k \) signifies the rate of interest, and \( t \) represents the time in years for which the interest is compounded. Continuous compounding means the frequency of compounding is infinite, leading to maximum possible growth of the invested amount. Some key points to remember about continually compounding interest:

• It results in a higher return than simple or periodically compounded interest due to the constant compounding effect.
• This form of interest is particularly important in fields like finance to calculate future investment values.
• The formula involves exponential growth, which means any increases are proportionate to the current value, resulting in rapid growth over time.

Properties of Logarithms

Logarithms are powerful mathematical tools used to reverse exponential processes. Understanding their properties is crucial when solving equations involving exponential terms. Key properties of logarithms include:

• Product Rule: \( \log_b(xy) = \log_b x + \log_b y \) - The log of a product is the sum of the logs.
• Quotient Rule: \( \log_b(\frac{x}{y}) = \log_b x - \log_b y \) - The log of a quotient is the difference of the logs.
• Power Rule: \( \log_b(x^y) = y \cdot \log_b x \) - The log of a power is the exponent times the log of the base.
• Change of Base Formula: \( \log_b x = \frac{\log_k x}{\log_k b} \) - Allows changing the base of a logarithm.

To solve the formula for continually compounding interest, you use the natural logarithm (\( \ln \)), where the base is \( e \). Applying the natural logarithm helps to isolate the variable \( t \) when you encounter the term \( e^{kt} \), using the fact that \( \ln(e^x) = x \). This property simplifies the equation and lets you solve for \( t \) in a straightforward way.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears as the exponent. They play a vital role in modeling growth processes, such as population growth, radioactive decay, and of course, continually compounding interest. In our context, the function \( y = A e^{kt} \) is an exponential function. Here, \( e \) is the base, which is found in many natural growth and decay processes. Exponential functions have several important characteristics:

• They are always positive, as the exponential of any real number is positive.
• They increase rapidly, provided their base is greater than one, reflecting continuous growth.
• Asymptotically approach zero for large negative inputs; hence, never actually reach zero unless \( A = 0 \).

These features make exponential functions suitable for describing processes that accelerate over time, such as continually compounding interest. Additionally, in these functions, small changes in the independent variable (\( t \) in our formula) can lead to large changes in the dependent variable (\( y \)), illustrating their sensitivity and power in growth modeling. Understanding how to manipulate and solve exponential functions is key to harnessing their potential in practical applications.