Problem 370 - Precalculus

Question

Recall the compound interest formula \(A=a\left(1+\frac{r}{k}\right)^{k t} .\) Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\)

Step-by-Step Solution

Answer

Time \(t\) is given by \(t = \frac{\ln\left(\frac{A}{a}\right)}{k \ln\left(1 + \frac{r}{k}\right)}\).

Step 1: Understand the Compound Interest Formula

We start with the compound interest formula: \(A = a\left(1 + \frac{r}{k}\right)^{kt}\), where \(A\) is the final amount, \(a\) is the principal amount, \(r\) is the annual interest rate, \(k\) is the number of times interest is compounded per year, and \(t\) is the time in years.

Step 2: Isolate the Exponential Expression

To solve for \(t\), first divide both sides by the principal amount \(a\): \(\frac{A}{a} = \left(1 + \frac{r}{k}\right)^{kt}\). This isolates the exponential expression.

Step 3: Take the Natural Logarithm of Both Sides

To bring down the exponent \(kt\), take the natural logarithm (ln) of both sides: \(\ln\left(\frac{A}{a}\right) = \ln\left( \left(1 + \frac{r}{k}\right)^{kt} \right)\).

Step 4: Apply Logarithm Power Rule

Use the logarithm power rule \(\ln(b^x) = x \ln(b)\) to simplify the right side: \(\ln\left(\frac{A}{a}\right) = kt \cdot \ln\left(1 + \frac{r}{k}\right)\).

Step 5: Solve for Time \(t\)

Finally, solve for \(t\) by dividing both sides by \(k \ln\left(1 + \frac{r}{k}\right)\): \(t = \frac{\ln\left(\frac{A}{a}\right)}{k \ln\left(1 + \frac{r}{k}\right)}\). This is the expression for time \(t\) as a function of the other variables.

Key Concepts

Understanding Logarithms

Logarithms are a fundamental mathematical concept that helps us work with exponential equations. At their core, logarithms are the inverse of exponentiation. This means if we have an equation of the form \(b^x = y\), then \(\log_b(y) = x\). In simple terms, a logarithm tells us the power to which a base number must be raised to get another number.
Here's a quick breakdown of some key points about logarithms:

• Base: The number that is raised to a power. Common bases are 10 and \(e\) (natural logarithms).
• The Algebra: \(\log_b(mn) = \log_b(m) + \log_b(n)\) means the log of a product is the sum of the logs.
• The Power Rule: \(\log_b(m^n) = n \cdot \log_b(m)\), which lets us deal with exponents more easily.

In the context of the compound interest problem, we used logarithms to bring down the exponent to solve for time \(t\). By understanding logarithms, we can transform and resolve exponential relationships.

Exponential Equations in Compound Interest

Exponential equations often appear in financial mathematics, such as with compound interest calculations. They involve expressions where the variable of interest is an exponent. In the compound interest formula, the equation \(A = a\left(1 + \frac{r}{k}\right)^{kt}\) is an example of an exponential equation.Understanding exponential equations is crucial because they describe how an amount grows over time. Here's what happens in an exponential equation:

• The base, \(1 + \frac{r}{k}\), represents the growth factor.
• The exponent \(kt\) determines how often and over what period the growth factor is applied.

When solving for an unknown variable, like time \(t\), dealing with exponential equations often requires the use of logarithms. This helps us "unwrap" the exponent from the base so that we can solve for \(t\) or any other unknown variable.

Interest Rate Calculation

Calculating the interest rate is a pivotal step in understanding how your investment grows over time. The interest rate \(r\) reflects how much profit is expected from an investment. In the compound interest formula \(A = a\left(1 + \frac{r}{k}\right)^{kt}\), \(r\) is the yearly interest and is often expressed as a decimal (for example, 5% becomes 0.05).To accurately calculate with interest rates, consider:

• Annual Rate: It's the percentage increase over a year and must be adjusted for smaller compounding intervals.
• Divisions: If interest compounds more frequently than annually, divide \(r\) by \(k\), the number of compounding periods per year.
• Growth Factor: Putting the adjusted rate into the formula shows how your investment grows over each period.

Understanding how to manipulate and calculate the interest rate is vital for effective financial planning and will show how your investment could perform over its life.

Time in Compound Interest

Determining the time \(t\) required for a particular investment goal using compound interest can be quite insightful. In the compound interest formula \(A = a\left(1+\frac{r}{k}\right)^{kt}\), \(t\) is the number of years it will take to reach a certain amount \(A\) from an initial principal investment \(a\).To solve for \(t\), we use the formula derived from manipulating the compound interest equation:\[ t = \frac{\ln(A/a)}{k \ln(1 + r/k)} \]Key considerations when solving for \(t\) include:

• Doubling Time: This is a common calculation where you determine how long it takes for an investment to double.
• Investment Goal: Knowing \(A\) and how much you want to invest helps isolate the necessary time \(t\).
• Frequency of Compounding: More frequent compounding can significantly affect the time \(t\) required to reach \(A\).

Mastering the calculation of time in compound interest is an essential skill for anyone looking to maximize the benefits of their investments over time.