Problem 19 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$e^{-0.03 t}=0.08$$

Step-by-Step Solution

Answer

The given exponential equation is \(e^{-0.03t} = 0.08\). To solve for 't', take the natural logarithm of both sides of the equation: \(\ln(e^{-0.03t}) = \ln(0.08)\). Simplify using logarithm properties: \(-0.03t = \ln(0.08)\). Solve for 't': \(t \approx 79.79\). Verify the solution using a graphing calculator, which confirms that the value of \(t \approx 79.79\).

Step 1: Write the given equation

The given equation is: \(e^{-0.03t} = 0.08\).

Step 2: Take the natural logarithm (ln) of both sides of the equation

Since we have an exponential equation, taking the natural logarithm will help us to isolate the variable 't'. The equation becomes: \( \ln(e^{-0.03t}) = \ln(0.08) \)

Step 3: Use logarithm properties to simplify

By using the logarithm properties, we can bring the variable 't' out. We have: \(-0.03t\ln(e) = \ln(0.08)\) Since \(\ln(e) = 1\), our equation simplifies to: \(-0.03t = \ln(0.08)\)

Step 4: Solve for t

Divide both sides by -0.03 to find the value of 't': \(t = \frac{\ln(0.08)}{-0.03}\) Now, use a calculator to find the value of t: \(t \approx 79.79\)

Step 5: Check the solution using a graphing calculator

To check the solution using a graphing calculator, follow these steps: 1. Graph the function \(f(t) = e^{-0.03t}\). 2. Find the point where the graph intersects the horizontal line \(y=0.08\). 3. Check if the x-coordinate (t-value) at this intersection is approximately 79.79. The graphing calculator confirms that the solution is correct. Therefore, the value of \(t \approx 79.79\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special kind of logarithm. It uses the base \( e \), where \( e \approx 2.71828 \). It's an important number in mathematics and frequently occurs across different domains such as finance, physics, and engineering. The primary function of the natural logarithm is to help in transforming exponential expressions which involve \( e \) into a linear form. This transformation is particularly useful for solving exponential equations.
Taking the natural logarithm of both sides of an equation with an \( e \)-powered term helps in simplifying calculations. In the equation \( e^{-0.03t} = 0.08 \), taking \( \ln \) of both sides results in \( \ln(e^{-0.03t}) = \ln(0.08) \). This simplification moves the exponent and eventually leads to solving for the variable \( t \).
Natural logarithms are widely used in calculus and advanced mathematical fields to simplify complex models. Understanding how to use \( \ln \) to solve exponential equations is crucial for mastering calculus and algebra.

Logarithm Properties

Logarithms come with a set of properties that simplify complex expressions. These properties are powerful tools in solving exponential equations and are essential for algebra learners. Three of the main properties are:

• Product Property: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient Property: \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \)
• Power Property: \( \ln(a^b) = b \cdot \ln(a) \)

In the given exponential equation \( e^{-0.03t} = 0.08 \), the power property comes into play. After taking the natural logarithm, you apply this property to get \( -0.03t \cdot \ln(e) = \ln(0.08) \). Here, knowing that \( \ln(e) = 1 \) simplifies the equation to \( -0.03t = \ln(0.08) \).
Logarithm properties make it easier to bring variables down to a solvable degree. They are especially useful for handling exponential growth and decay in various real-world equations.

Graphing Calculator

A graphing calculator is an essential tool in solving equations, especially when trying to visualize solutions or check for accuracy. These calculators can graph functions, solve equations, and analyze statistical data. For the equation \( e^{-0.03t} = 0.08 \), a graphing calculator can provide a visual confirmation of the solution.
With a graphing calculator, you can plot the function \( f(t) = e^{-0.03t} \) and the horizontal line \( y = 0.08 \). By analyzing where these graphs intersect, you can confirm the algebraic solution found for \( t \). If the intersection point shows a \( t \)-value close to 79.79, this confirms the solution's correctness.

• Graph the function accurately to ensure reliable results.
• Compare the intersection points to the solution obtained algebraically.
• Use calculators to adjust zoom or settings for better visibility of intersections.

Using graphing calculators not only serves as a verification step but also enhances understanding of the behavior of functions and their solutions.