Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$1000 e^{-4 x}=75$$

Step-by-Step Solution

Answer

The solution to the equation \(1000 e^{-4 x} = 75\) round up to three decimal places is approximately \(x = 2.200\), as confirmed by plotted graph.

Step 1: Simplify The Equation

Divide both sides of the equation by 1000 to simplify the equation to be: \(e^{-4x} = 0.075\)

Step 2: Apply A Logarithm To Both Sides

Apply a natural log (ln) to both sides to remove the exponent on e. Due to the properties of logarithms, the equation becomes: \(-4x = \ln(0.075)\)

Step 3: Solve For Variable x

Divide both sides of the equation by -4 to isolate \(x\). The equation simplifies to: \(x = -\frac{1}{4} \ln(0.075)\)

Step 4: Calculate The Value Of x

Calculate the numerical value of \(x\) by substituting \(\ln(0.075)\) into the equation. Round the final answer to three decimal places.

Step 5: Verify The Solution With Graphing Utility

Plot the function \(f(x) = 1000e^{-4x}\) and the line \(y = 75\). The x-coordinate of the point of intersection confirms the solution obtained algebraically.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are widely used to solve exponential equations when the variable is in the exponent. The general form is expressed as \( y = \log_b(x) \) where \( b \) is the base and \( x \) is the argument of the logarithm. The logarithm of a number is the power to which the base must be raised to obtain that number.

For the equation \( f(x) = 1000e^{-4x} \) where \( f(x) = 75 \) from the example, the first step involves isolating the exponential expression by dividing both sides by 1000, leading to \( e^{-4x} = 0.075 \). To solve for \( x \) when it is in the exponent, you apply the logarithmic function to both sides. This practice allows you to bring down the exponent and solve for the variable in a more straightforward linear fashion.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \) where \( e \) is an irrational constant approximately equal to 2.71828. It's often used in calculus and the solutions of exponential equations involving the base \( e \).

In the given exercise, once we have the exponential equation \( e^{-4x} = 0.075 \) from Step 1, applying a natural log to both sides facilitates the removal of the base \( e \) since \( \ln(e^x) = x \). This step simplifies the equation, utilizing the natural logarithm's ability to 'unwind' the exponential function, making it possible to solve for the variable.

Properties of Logarithms

Understanding the properties of logarithms is crucial in manipulating and solving logarithmic equations. Some essential properties include:

• Product Property: \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \)
• Quotient Property: \( \log_b(x / y) = \log_b(x) - \log_b(y) \)
• Power Property: \( \log_b(x^y) = y \cdot \log_b(x) \)

In Step 2 of our solution, we use the property that \( \ln(e^x) = x \) to simplify \( e^{-4x} \). This results in the equation \( -4x = \ln(0.075) \), demonstrating how logarithmic properties can help in isolating the variable.

Graphing Utility Verification

After solving the equation algebraically, it's often helpful to verify the solution graphically, especially for complex functions. With graphing utilities, such as graphing calculators or software, you can plot the original function and a constant function representing the other side of the equation.

The algebraic solution provided the value of \( x \) after the logarithmic manipulation of the given exponential equation. The final step is to verify this solution graphically. By plotting \( f(x) = 1000e^{-4x} \) and \( y = 75 \) on a graph, the point where these two graphs intersect is the solution to the equation. In our exercise, if the x-coordinate of the intersection point matches the algebraically found solution (within a suitable tolerance), it validates the result.