Problem 80 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Goal

Our goal is to solve the compound interest formula \(A = a\left(1+\frac{r}{k}\right)^{kt}\) for the variable \(t\). To do this, we will use logarithmic properties to isolate \(t\).

Step 2: Apply Logarithm to Both Sides

To isolate the exponent containing \(t\), take the natural logarithm (or log to any base) of both sides: \[\ln(A) = \ln\left(a\left(1+\frac{r}{k}\right)^{kt}\right)\]

Step 3: Use the Power Property of Logarithms

Apply the power property of logarithms to bring the exponent down:\[\ln(A) = \ln(a) + kt\cdot \ln\left(1+\frac{r}{k}\right)\] This uses the property \(\ln(x^y) = y\cdot \ln(x)\).

Step 4: Isolate the Term Containing \(t\)

Subtract \(\ln(a)\) from both sides to focus on terms containing \(t\):\[\ln(A) - \ln(a) = kt\cdot \ln\left(1+\frac{r}{k}\right)\]

Step 5: Solve for \(t\)

Divide both sides by \(k\cdot \ln\left(1+\frac{r}{k}\right)\) to solve for \(t\):\[t = \frac{\ln(A) - \ln(a)}{k \cdot \ln\left(1+\frac{r}{k}\right)}\] This isolates \(t\) on one side of the equation.

Key Concepts

Logarithms

Logarithms are incredibly useful when dealing with exponential equations, like the one used in calculating compound interest. Imagine you have a number and you want to know, "To what power must I raise a particular base to get that number?" That's essentially what logarithms tell you. For example, if you have an exponentiation where the base is 10, the logarithm helps you find the power to which 10 must be raised to result in a particular number.
Logarithms have several key properties that make them handy in mathematical equations:

• Power Property: If you have \(\log_b(x^y) = y \cdot \log_b(x)\), this allows you to bring down the exponent in front of the logarithm, simplifying calculations.
• Product Property: The logarithm of a product is the sum of the logarithms of the factors: \(\log_b(xy) = \log_b(x) + \log_b(y)\).
• Quotient Property: Similarly, \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\).

By using these properties, you can manipulate and simplify complex expressions, making logarithms an essential tool for solving equations involving exponential growth or decay.

Exponential Equations

Exponential equations occur when variables appear as exponents. In the context of compound interest, you're dealing with an equation like \(A = a\left(1+\frac{r}{k}\right)^{kt}\), where the term \( (1+\frac{r}{k})^{kt} \) represents exponential growth of the principal amount with interest.
When solving exponential equations:

• First, isolate the exponential expression on one side of the equation.
• Apply logarithms to both sides. By doing so, you convert the exponential equation into a linear form, which is easier to solve.
• The power property of logarithms is particularly beneficial here. It allows the exponent to be moved down in front of the logarithm, effectively transforming a multiplication problem into an addition or subtraction problem that is much more manageable.

Once the equation is linear, it becomes straightforward to isolate and solve for the unknown variable. This technique is widely used not just in financial calculations but in various scientific domains involving exponential growth and decay.

Financial Mathematics

In financial mathematics, compound interest is a fundamental concept that describes how an investment grows over time. The formula \(A = a\left(1+\frac{r}{k}\right)^{kt}\) is central to understanding compound interest. Here, \(A\) represents the future value of the investment, \(a\) is the initial principal balance, \(r\) is the annual interest rate, \(k\) is the number of times interest is compounded per year, and \(t\) is the time in years.
The magic of compound interest lies in its ability to grow your investment at a faster rate, as interest is earned on both the initial principal and the accumulated interest from previous periods. To solve for time \(t\), you often need to rearrange the formula, which involves understanding and applying logarithms and exponential equations. This is key to planning investments, loans, and any financial scenarios involving growth or accumulation over time.Some key points to remember:

• Increasing the compounding frequency (higher \(k\)) generally increases the accumulated amount due to more frequent interest calculations.
• The compound interest formula allows for predicting the future value of an investment or the time required to reach a financial goal.
• Understanding the impact of each variable can help in making informed financial decisions.

Financial mathematics empowers investors, helping them maximize returns and strategically plan for the future.