Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use logarithms to solve each problem. How long will it take an investment of $$\$ 8000$$ to double if the investment earns interest at the rate of \(8 \%\) compounded continuously?

Step-by-Step Solution

Answer

It will take approximately \(8.66\) years for the investment to double.

Step 1: Write down the continuous compounding formula

First, we need to write down the formula for continuous compounding: \[A = P * e^{rt}\]

Step 2: Set the values of A, P, and r

Next, we will substitute the given values of \(A, P,\) and \(r\) into the formula. We have \(A = 16000\), \(P = 8000\), and \(r = 0.08\). Plug these values into the formula: \[16000 = 8000 * e^{0.08t}\]

Step 3: Divide both sides by 8000

Now, to solve for \(t\), let's first divide both sides of the equation by 8000: \[\frac{16000}{8000} = \frac{8000 * e^{0.08t}}{8000}\] \[2 = e^{0.08t}\]

Step 4: Take the natural logarithm of both sides

Now we will take the natural logarithm (base \(e\)) of both sides to solve for \(t\): \[\ln(2) = \ln(e^{0.08t})\]

Step 5: Use the logarithm property to bring out the exponent

Using the logarithm property \(\ln(a^b) = b\ln(a)\), we can bring the exponent out: \[\ln(2) = 0.08t\ln(e)\] Since \(\ln(e) = 1\), the equation simplifies to: \[\ln(2) = 0.08t\]

Step 6: Solve for t

Finally, to find the value of \(t\), we can divide both sides of the equation by 0.08: \[t = \frac{\ln(2)}{0.08}\] Now, use a calculator to compute the value: \[t \approx \frac{0.6931}{0.08}\] \[t \approx 8.66\] So, it will take approximately \(8.66\) years for the investment to double.

Key Concepts

Logarithms in Finance

Logarithms are incredibly useful in the financial sector for a variety of applications, including the calculation of interest rates, compound interest, and growth rates. In our exercise, we utilize logarithms to determine the time it takes for an investment to double with continuous compounding interest.

Continuous compounding is a concept where interest is calculated an infinite number of times over a given period, as opposed to annually, monthly, or daily. This method leads to an exponential increase in the initial investment over time. To solve problems involving continuous compounding, we use the natural logarithm because the underlying equations involve the mathematical constant e, which is approximately equivalent to 2.71828.

To find the time needed for an investment to double, we first have to set up the problem using the basic continuous compounding formula, and then isolate the variable representing time (t). This involves dividing both sides of the equation to simplify it and then applying the properties of logarithms to solve for t. Finding how long it takes for investments to grow is essential for financial planning and investment strategies.

Natural Logarithm

The natural logarithm is a logarithm with base e, where e is an irrational constant approximately equal to 2.71828. It is denoted by ln and is especially important in dealing with problems of continuous growth or decay, like our investment problem.

In step 4 of our solution process, we employ the natural logarithm to both sides of the equation in order to remove the exponential function involving e. The unique relationship between e and its logarithm means that the natural logarithm of e to any power is simply that power—this is why (e)e^x simplifies to x.

Understanding the properties of the natural logarithm is critical when working with continuous compounding formulas. It also comes in handy in many areas of finance, economics, and even in natural sciences when we describe phenomena that exhibit continuous growth rates.

Time Value of Money

The time value of money (TVM) is a financial concept suggesting that money available now is worth more than the same amount in the future due to its potential earning capacity. This fundamental idea underpins the concept of interest and the basis for investments growing over time.

With continuous compounding, we capitalize on the TVM by earning interest upon interest in an exponential manner. The formula we used in the exercise, A = Pe^(rt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, and t is the time the money is invested for, serves to calculate precisely how much value we can expect our money to gain over time.

By understanding the TVM, individuals can make more informed decisions about saving, investing, and borrowing. Knowing how to compute the time needed for an investment to reach a certain value is a practical application of the TVM that helps investors in their financial forecasting and planning.