Problem 80 - College Algebra with Corequisite Support

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

The solution for time \(t\) is \(t = \frac{\ln\left(\frac{A}{a}\right)}{k \cdot \ln\left(1+\frac{r}{k}\right)}\)."

Step 1: Understand the Problem

We have the formula for compound interest: \(A = a\left(1+\frac{r}{k}\right)^{kt}\), where \(A\) is the final amount, \(a\) is the initial principal, \(r\) is the annual interest rate, \(k\) is the number of times interest is compounded per year, and \(t\) is the time in years. Our task is to solve this equation for \(t\).

Step 2: Divide by the Initial Principal

To start isolating \(t\), divide both sides of the equation by \(a\) to get \(\frac{A}{a} = \left(1+\frac{r}{k}\right)^{kt}\). This simplifies the equation and prepares it for logarithms.

Step 3: Apply Logarithms

Take the natural logarithm of both sides. This gives us \(\ln\left(\frac{A}{a}\right) = \ln\left(\left(1+\frac{r}{k}\right)^{kt}\right)\). Using logarithm properties, \(\ln(b^x) = x\ln(b)\), the equation becomes \(\ln\left(\frac{A}{a}\right) = kt \cdot \ln\left(1+\frac{r}{k}\right)\).

Step 4: Solve for Time \(t\)

Isolate \(t\) by dividing both sides of the equation by \(k \cdot \ln\left(1+\frac{r}{k}\right)\), giving us \(t = \frac{\ln\left(\frac{A}{a}\right)}{k \cdot \ln\left(1+\frac{r}{k}\right)}\). This expression gives the time \(t\) in terms of the other variables in the compound interest formula.

Key Concepts

Logarithmic Properties

Logarithms are a mathematical tool that can help simplify equations, especially when dealing with exponential terms. One key property of logarithms is how they can transform a product or power into a sum or a multiplication. For example, the property \( \ln(b^x) = x \cdot \ln(b) \) allows us to deal with the exponent by bringing it in front of the logarithm. This property is particularly useful in solving the compound interest formula for time because it transforms the problem into a more manageable linear form.

In the context of the compound interest equation, taking the natural logarithm of both sides:

• Enables the exponent involving time \( t \) to be extracted.
• Simplifies the expression \( \ln\left((1 + \frac{r}{k})^{kt}\right) = kt \cdot \ln\left(1 + \frac{r}{k}\right) \).
• Prepares the equation for isolating \( t \) and finding its value in terms of other known quantities.

Understanding these logarithmic properties is crucial in manipulating the formula and solving real-world problems like determining how long an investment needs to grow to reach a certain value.

Mathematical Equations

In mathematical equations, especially those involving compound interest, the goal is often to solve for an unknown variable. With the compound interest equation, we specifically aim to solve for the time variable \( t \). Here's how we systematically break it down:

The compound interest formula is \(A = a\left(1+\frac{r}{k}\right)^{kt}\). Our first step is to isolate \( t \), hence,

• We divide both sides of the equation by the initial amount \( a \) yielding \( \frac{A}{a} = \left(1+\frac{r}{k}\right)^{kt} \).
• Next, by applying logarithms, we transform the equation so that \( kt \) becomes a coefficient, facilitating the next step of isolating \( t \).
• We then divide through by \( k \cdot \ln(1 + \frac{r}{k}) \) to solve for \( t \), resulting in \( t = \frac{\ln\left(\frac{A}{a}\right)}{k \cdot \ln(1 + \frac{r}{k})} \).

These steps show how basic algebra along with logarithmic properties allow us to derive solutions for complex problems like compound interest with elegance and simplicity.

Time Calculation

Calculating the time period in a compound interest setting can seem daunting, but with the right approach, it becomes manageable. In these scenarios, the time \( t \) signifies how long it will take for an investment to grow from its initial value to some future amount given the compounding factors.

When we solved for \( t \) using

• The logarithmic manipulation, we've effectively determined the number of years required for the initial principal \( a \) to reach \( A \).
• This is particularly useful in situations where you need to make financial predictions or decisions based on investment growth.
• It allows for strategic planning, as knowing the time to achieve a financial goal helps in formulating investment strategies and assessing the risk.

Practically, this calculation helps individuals and businesses alike in assessing how their funds will grow over time, ensuring that their financial plans are on track to meet their objectives. Whether you're aiming for retirement savings or calculating loan repayment schedules, understanding how to compute time in compound interest scenarios is crucial.