Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. $$r=p-k \ln t, \text { for } t$$

Step-by-Step Solution

Answer

The solution for \(t\) is \(t = e^{\frac{r-p}{k}}\).

Step 1: Isolate the logarithmic term

To begin solving for the variable \(t\), we first need to isolate the term containing \(t\) on one side of the equation. We will rearrange the equation by subtracting \(p\) and then adding \(k \ln t\) to both sides, resulting in \(p - r = -k \ln t\).

Step 2: Eliminate the negative sign

To clear the negative sign on the right side of the equation, multiply both sides by -1. This gives us \(r - p = k \ln t\).

Step 3: Solve for the logarithmic term

Now, isolate the \(\ln t\) term by dividing both sides of the equation by \(k\). This results in \(\ln t = \frac{r - p}{k}\).

Step 4: Exponentiate both sides

To solve for \(t\), we need to eliminate the natural logarithm by applying the exponential function to both sides. This gives us \(t = e^{\frac{r-p}{k}}\).

Key Concepts

Understanding Logarithms

Logarithms are an essential tool in solving various mathematical problems, particularly when dealing with exponential growth or decay. The logarithm of a number is the exponent to which another fixed number, the base, is raised to produce that number. The formula for the logarithm is:

• Logarithmic form: If \( b^x = y \), then \( \log_b(y) = x \).
• Natural Logarithm: It is a logarithm with base \( e \), where \( e \approx 2.718 \). Notated as \( \ln x \).

In our exercise, \( \ln t \) represents the natural logarithm of \( t \). This step is important because it helps transform multiplicative relationships into additive ones, which are easier to solve. To isolate \( \ln t \), basic algebraic manipulation is employed, and then the natural logarithm is reversed using exponentiation.

Working with Exponential Functions

Exponential functions are functions where the variable is an exponent. They take the form of \( y = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm. These functions are characterized by rapid growth or decay.

• When exponentiating both sides of the equation, you apply the operation of raising \( e \) to the power of both sides of \( \ln t = \frac{r - p}{k} \).
• This results in \( t = e^{\frac{r-p}{k}} \).

Exponentiation is key here because it effectively cancels out the natural logarithm, leaving us with a simple expression for \( t \). This process showcases how logarithmic and exponential functions are inverses of each other, used in tandem to solve equations.

Mastering Algebraic Manipulation

Algebraic manipulation is a fundamental skill in mathematics that involves rearranging equations to isolate desired variables. This involves utilizing arithmetic operations such as addition, subtraction, multiplication, and division.

• Step 1: Rearrange the equation to separate the term with \( t \), moving \( p \) to one side and the \( k \ln t \) term to the other, leading to \( p - r = -k \ln t \).
• Step 2: Address the negative sign by multiplying the entire equation by \(-1\), simplifying it to \( r - p = k \ln t \).
• Step 3: Divide by \( k \) to isolate \( \ln t \), resulting in \( \ln t = \frac{r-p}{k} \).

Each step methodically focuses on reducing the equation to its simplest form, allowing us to effectively find the value of \( t \). This ensures that complex equations become manageable, demonstrating the power of algebraic manipulation in solving equations.