Problem 79 - College Algebra

Question

Recall the formula for continually compounding interest, \(y=A e^{k t} .\) Use the defin tion of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

Step-by-Step Solution

Answer

Time \(t\) is \(t = \frac{1}{k} \ln\left(\frac{y}{A}\right)\).

Step 1: Write Down the Given Formula

The formula for continually compounding interest is given as \(y = A e^{kt}\). Our goal is to solve for the time \(t\).

Step 2: Isolate the Exponential Term

Divide both sides of the equation by \(A\) to isolate the exponential term:\[\frac{y}{A} = e^{kt}\]

Step 3: Apply the Definition of a Logarithm

Take the natural logarithm of both sides to eliminate the exponential function. This is because the natural logarithm \(\ln\) is the inverse of the exponential function \(e^x\):\[\ln\left(\frac{y}{A}\right) = \ln(e^{kt})\]

Step 4: Simplify the Logarithm

Apply the property of logarithms \(\ln(e^{x}) = x\). Thus, the equation simplifies to:\[\ln\left(\frac{y}{A}\right) = kt\]

Step 5: Solve for Time \(t\)

To find \(t\), simply divide by \(k\):\[t = \frac{1}{k} \ln\left(\frac{y}{A}\right)\]

Step 6: Conclusion: Express \(t\) as a Single Logarithm

The time \(t\) is expressed using a single natural logarithm as:\[t = \ln\left(\frac{y}{A}\right)^{1/k}\]

Key Concepts

Exponential Functions

When we talk about exponential functions, we refer to expressions where a constant base is raised to a variable exponent. In the formula for continually compounding interest, the expression \( e^{kt} \) represents an exponential function. Here, the base \( e \) is a mathematical constant approximately equal to 2.718, which is known as Euler's number.

Exponential functions are prevalent in real-world phenomena. They model growth or decay, such as population growth, radioactive decay, and, as in our exercise, compound interest. Here are some important characteristics you should remember about exponential functions:

• Growth or decay happens at a constant percentage rate relative to the current value.
• They continue increasing or decreasing rapidly over time.
• The graph of an exponential function is a smooth curve that either rises or falls steadily and does not intersect the x-axis.

Their indispensable role in various growth and decay models makes them crucial in mathematics and science.

Compound Interest

Compound interest is a fascinating and critical concept in finance where the interest earned over time is added back to the principal sum, and thereafter, the interest for subsequent periods is computed on this new principal. This is different from simple interest, where the interest is calculated only on the original principal.

The formula \( y = A e^{kt} \) reflects continuously compounded interest, meaning that the frequency of compounding is infinite. Such scenarios are idealized as they reflect how savings can grow rapidly over time when the interest is continuously compounded.

Key elements of the compound interest formula include:

• \( A \): The initial deposit or principal amount.
• \( y \): The total amount after interest.
• \( k \): The interest rate in decimal form.
• \( t \): The time for which the interest is calculated.

This formula helps to forecast how money can grow with time and is pivotal in personal finance, investments, and savings planning. Understanding this concept is significant for making informed financial decisions.

Natural Logarithms

Natural logarithms are essential in mathematics, particularly because they serve as the inverse of the exponential function. When dealing with the exponential equation \( y = A e^{kt} \), using natural logarithms allows us to solve for unknown variables like \( t \), capturing how long it takes for an investment to grow to a particular amount under compound interest.

The natural logarithm of a number, denoted as \( \ln(x) \), essentially gives us the power to which \( e \) must be raised to obtain \( x \). Here are some noteworthy points about natural logarithms:

• The property \( \ln(e^x) = x \) allows us to simplify the expressions where \( e \) is the base.
• They are particularly useful in conversion of multiplicative relationships into additive ones through their properties.
• Natural logarithms appear naturally within mathematics when dealing with continuous growth or decay processes.

By comprehensively understanding natural logarithms, you equip yourself with a valuable tool for analyzing exponential growth scenarios, such as those in financial calculations or physical processes.