Problem 68
Question
Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). Find a formula that gives the time needed for an investment of \(P\) dollars to double, if the interest rate is \(r \%\) compounded annually.
Step-by-Step Solution
Verified Answer
Question: Determine the time required for an investment to double if the interest rate is compounded annually.
Answer: The time needed for an investment to double, if the interest rate is compounded annually, can be found using the formula $t = \frac{\ln{2}}{\ln{(1+r)}}$, where $t$ is the time in years and $r$ is the interest rate.
1Step 1: Substitute the final amount with double the principal
Since we are finding the time needed for the investment to double, we can replace \(A\) with \(2P\). The equation becomes:
$$2P=P(1+r)^{t}$$
2Step 2: Divide both sides by the principal
Now, we will divide both sides of the equation by \(P\) to eliminate it. This results in:
$$2 = (1+r)^{t}$$
3Step 3: Apply the logarithm to both sides of the equation
In order to solve for \(t\), we can apply the natural logarithm to both sides of the equation:
$$\ln{2} = \ln{(1+r)^{t}}$$
4Step 4: Use the logarithm power rule
Using the logarithm power rule, we can simplify the expression on the right side of the equation. The power rule states that \(\ln{a^b} = b\ln{a}\). Applying this rule, we obtain:
$$\ln{2} = t\ln{(1+r)}$$
5Step 5: Solve for the time t
Finally, to find the formula for \(t\), we will isolate it by dividing both sides of the equation by \(\ln{(1+r)}\):
$$t = \frac{\ln{2}}{\ln{(1+r)}}$$
This formula gives the time needed for an investment of \(P\) dollars to double, if the interest rate is \(r\%\) compounded annually.
Key Concepts
Natural LogarithmExponential GrowthDoubling Time
Natural Logarithm
The natural logarithm is an essential mathematical function that often appears in problems involving exponential growth and compound interest. It's usually denoted as \( \ln \), and it represents the logarithm to the base \( e \), where \( e \approx 2.71828 \). In other words, if you see \( \ln{x} \), it's asking the question: "To what power must \( e \) be raised to get \( x \)?"
Natural logarithms are particularly useful when dealing with exponential functions since they help us inverting exponential terms easily. For instance, when you have an equation like \( e^y = x \), you can solve for \( y \) using the natural logarithm: \( y = \ln{x} \).
In the context of compound interest, the natural logarithm helps us transform equations that involve exponential expressions into a form where we can solve for time or other variables more straightforwardly. This transformation is critical when determining the "age" of an investment or comparing different growth scenarios.
Natural logarithms are particularly useful when dealing with exponential functions since they help us inverting exponential terms easily. For instance, when you have an equation like \( e^y = x \), you can solve for \( y \) using the natural logarithm: \( y = \ln{x} \).
In the context of compound interest, the natural logarithm helps us transform equations that involve exponential expressions into a form where we can solve for time or other variables more straightforwardly. This transformation is critical when determining the "age" of an investment or comparing different growth scenarios.
Exponential Growth
Exponential growth occurs when the growth rate of a value is proportional to its current size. This characteristic leads to accelerated growth over time, with changes occurring more and more rapidly. In financial contexts, exponential growth is a vital concept since it describes how investments increase in value when interest is compounded.
The general formula for exponential growth, often found in the context of compound interest, is \( A = P(1 + r)^t \). Here, \( A \) is the final amount, \( P \) is the initial principal, \( r \) is the rate of interest per period, and \( t \) is the time the money is invested or borrowed.
When it comes to financial growth over time, even small interest rates can lead to considerable increases due to the compounding effect. This exponential nature means that over long periods, an investment can grow significantly, making exponential growth an exciting yet manageable challenge in financial planning and analysis.
The general formula for exponential growth, often found in the context of compound interest, is \( A = P(1 + r)^t \). Here, \( A \) is the final amount, \( P \) is the initial principal, \( r \) is the rate of interest per period, and \( t \) is the time the money is invested or borrowed.
When it comes to financial growth over time, even small interest rates can lead to considerable increases due to the compounding effect. This exponential nature means that over long periods, an investment can grow significantly, making exponential growth an exciting yet manageable challenge in financial planning and analysis.
Doubling Time
Doubling time is the period it takes for a quantity to grow to twice its size at a constant rate of growth. In finance, it refers to how long it takes for an investment or amount of money to double when it earns interest that compounds.
To find the doubling time with a specific interest rate compounded annually, we use a derivation from the compound interest formula resulting in \( t = \frac{\ln{2}}{\ln{(1+r)}} \). This formula is derived by setting the final amount equal to twice the principal and involves solving for \( t \) using logarithms.
Some key points about doubling time:
To find the doubling time with a specific interest rate compounded annually, we use a derivation from the compound interest formula resulting in \( t = \frac{\ln{2}}{\ln{(1+r)}} \). This formula is derived by setting the final amount equal to twice the principal and involves solving for \( t \) using logarithms.
Some key points about doubling time:
- It's a concise way to understand the effect of compound interest at a glance.
- Lower interest rates mean longer doubling times, while higher rates shorten it.
- Doubling time provides an intuitive way to compare investment opportunities.
Other exercises in this chapter
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