Problem 13 - College Algebra with Corequisite Support

Question

A logarithmic model is given by the equation \(h(p)=67.682-5.792 \ln (p) .\) To the nearest hundredth, for what value of \(p\) does \(h(p)=62 ?\)

Step-by-Step Solution

Answer

\( p \approx 2.67 \).

Step 1: Substitute and prepare

In the given model, we're looking for the value of \( p \) such that \( h(p) = 62 \). Start by substituting 62 for \( h(p) \) in the equation: \[ 62 = 67.682 - 5.792 \ln(p) \]

Step 2: Isolate the logarithm

Rearrange the equation to isolate the logarithmic term. Subtract 67.682 from both sides of the equation:\[ 62 - 67.682 = -5.792 \ln(p) \] Calculate the left-hand side: \[ -5.682 = -5.792 \ln(p) \].

Step 3: Simplify the equation

Divide both sides by -5.792 to solve for \( \ln(p) \): \[ \ln(p) = \frac{-5.682}{-5.792} \] Now simplify: \[ \ln(p) = 0.9809 \].

Step 4: Solve for p

To find \( p \), exponentiate both sides to eliminate the natural logarithm. Use the exponential function: \[ p = e^{0.9809} \].

Step 5: Calculate the value of p

Compute the value of \( p \) using a calculator: \[ p \approx e^{0.9809} \approx 2.667 \].

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a logarithm to the base of the mathematical constant \( e \), where \( e \) is approximately 2.71828. The natural logarithm of a number answers the question: "To what power must \( e \) be raised to obtain this number?" For example, \( \ln(e) = 1 \) because \( e^1 = e \). This type of logarithm is prevalent in mathematical equations, often found in calculus and continuous growth models.

• The natural logarithm is also the inverse of the exponential function, meaning \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).
• It's particularly useful for solving equations involving exponential growth and natural decay due to its unique properties related to the number \( e \).
• Common properties include: \( \ln(xy) = \ln(x) + \ln(y) \) and \( \ln(x/y) = \ln(x) - \ln(y) \).

Understanding the natural logarithm is essential for dealing with equations like in the original exercise, where we apply it to reverse the effects of an exponent.

Exponential Function

The exponential function is a mathematical function that grows rapidly as its input value increases. It is expressed as \( e^x \), where \( e \) is Euler's number. This function is significant due to its constant growth rate, which is proportional to its value at every point.

• Exponential functions appear in many real-world scenarios, such as population growth, radioactive decay, and interest calculations.
• They are defined mathematically as: \( f(x) = e^x \), where every increase by 1 in the variable \( x \) multiplies the function’s current value by \( e \).
• The exponential function is the inverse of the natural logarithm, meaning if \( y = e^x \), then \( x = \ln(y) \).

Using the exponential function, as seen in the final step of the exercise, allows us to solve for \( p \) by reversing the natural logarithm.

Solving Equations

Equation solving involves finding unknown values that satisfy a mathematical statement. For the given logarithmic model, we are solving for the variable \( p \). The process requires manipulating the equation step-by-step to isolate \( p \).

• First, substitute values into the equation, then rearrange terms to isolate the logarithmic part, as seen in the steps where \( 67.682 \) is subtracted from both sides.
• The next step involves simplifying the equation to solve for \( \ln(p) \), and then we use the properties of logarithms to substitute \( \ln(p) \) back into the equation.
• Finally, utilize the exponential function to solve for \( p \) by converting the logarithmic form back to its exponential equivalent, thus finding the approximate value of \( p \).

Solving such equations requires a basic understanding of how logarithms and exponential functions interact, providing a systematic approach to isolating the desired variable.