Problem 69 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \( t \) is \( e^{\frac{p-r}{k}} \).

Step 1: Isolate the Logarithm Term

Start with the given equation: \( r = p - k \ln t \). To solve for \( t \), we first need to isolate the logarithm term. Add \( k \ln t \) to both sides and subtract \( r \) from both sides to get: \( k \ln t = p - r \).

Step 2: Divide by k

Now, divide both sides of the equation by \( k \) to solve for \( \ln t \): \( \ln t = \frac{p - r}{k} \). This simplifies the equation and isolates \( \ln t \) on one side.

Step 3: Exponentiate to Solve for t

To eliminate the natural logarithm \( \ln \), exponentiate both sides of the equation using the base of the natural logarithm, which is \( e \). This gives: \( t = e^{\frac{p - r}{k}} \). Now, \( t \) is isolated and expressed in terms of \( p, r, \) and \( k \).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms, which are the inverse operations of exponentials. They help us solve equations where the unknown appears as the argument of a logarithm. In the equation \( r = p - k \ln t \), the logarithm \( \ln t \) can be thought of as saying, "To what power must \( e \) be raised to obtain \( t \)?" Whenever dealing with logarithmic equations, the goal is often to isolate the logarithm so that it can be inverted using exponentiation. This transforms the logarithmic form into a standard arithmetic form that is easier to manipulate for isolating variables.

Isolating Variables

Isolating variables is a crucial step in algebra, especially when dealing with complex equations. It involves manipulating the equation to express a particular variable explicitly on one side of the equality. In our problem, to solve for \( t \), we first aimed to isolate the term \( \ln t \). We did this by rearranging the equation \( r = p - k \ln t \) into \( k \ln t = p - r \). Knowing how to strategically move terms across the equals sign, while maintaining equality, is fundamental. This often includes adding, subtracting, multiplying, or dividing both sides of the equation by the same number so that the target variable is alone on one side. This makes it simpler to solve for the desired variable or expression.

Exponential Functions

Exponential functions are a key concept in mathematics and are the opposites of logarithmic functions. They take the form \( a^x \), where \( a \) is the base and \( x \) is the exponent. In the context of solving \( \ln t = \frac{p - r}{k} \), we use exponentiation to "undo" the logarithm, helping to reveal the variable \( t \). Specifically, using the base \( e \), we applied \( e ^{\ln t} = t \), resulting in \( t = e^{\frac{p - r}{k}} \). This step demonstrates how exponential functions can convert the natural logarithm into a straightforward algebraic expression, allowing us to solve for the variable \( t \). Understanding this relationship between exponentials and logarithms is essential for solving equations that involve these types of expressions.