Problem 18 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$e^{-t}=0.04$$

Step-by-Step Solution

Answer

The given exponential equation is \(e^{-t}=0.04\). By taking the natural logarithm of both sides, we obtain \(-t = ln(0.04)\). Then, we solve for t as \(t = -ln(0.04)\) and find that \(t \approx 3.22\). To verify our solution, we use a graphing calculator to plot the function \(y = e^{-t}\) and check the x-coordinate at the intersection of the horizontal line at \(y = 0.04\), which should be approximately equal to 3.22.

Step 1: Write down the given equation

Given equation: \(e^{-t} = 0.04\)

Step 2: Take the natural logarithm of both sides

Apply ln to both sides of the equation: \(ln(e^{-t}) = ln(0.04)\)

Step 3: Use properties of logarithms

Using the property that \( ln(a^b) = b*ln(a) \), we can simplify the equation: \(-t * ln(e) = ln(0.04)\) Since \(ln(e) = 1\), the equation simplifies to: \(-t = ln(0.04)\)

Step 4: Solve for t

To solve for t, multiply both sides of the equation by -1: \(t = -ln(0.04)\) Use a calculator to find the numerical value of t: \(t \approx 3.22\)

Step 5: Check the solution using a graphing calculator

To check the solution: 1. Graph the equation \(y = e^{-t}\) on the graphing calculator. 2. Find the point where the graph intersects the horizontal line representing \(y = 0.04\). 3. Check that the x-coordinate of this point is approximately equal to the value of t we found (3.22). If the x-coordinate of the intersection point is approximately equal to 3.22, our solution is correct.

Key Concepts

Natural Logarithms

Natural logarithms are a special type of logarithm with the base of the mathematical constant \( e \), which is approximately equal to 2.71828. The natural logarithm is often denoted by \( \ln(x) \) and plays a crucial role in solving exponential equations. In the context of the equation \( e^{-t} = 0.04 \), taking the natural logarithm of both sides helps us to bring the exponent down and simplify the equation into a more manageable form. Some important properties of natural logarithms include:

• The natural logarithm of \( e \) itself is 1, since \( \ln(e) = 1 \).
• Natural logarithms allow us to transform multiplicative relationships into additive ones, making complex equations easier to handle.

To use natural logarithms effectively, remember these characteristics as they're fundamental in algebraic manipulations needed to solve equations like the one we are considering.

Graphing Calculators

Graphing calculators are powerful tools used to visualize mathematical equations and confirm algebraic solutions. To check the solution for \( e^{-t} = 0.04 \), the graphing calculator can graph the function \( y = e^{-t} \) and a horizontal line \( y = 0.04 \).Here are the general steps to check solutions using a graphing calculator:

• Input the function into the graphing calculator.
• Set a viewing window that encompasses the expected solution using a good range for x-values, in this case around the approximate value 3.22.
• Analyze the intersection point where the graph of \( y = e^{-t} \) meets the line \( y = 0.04 \).
• Verify that the x-coordinate of this intersection matches the algebraic solution.

The graphing calculator is an excellent way to visually confirm your algebraic work, ensuring both accuracy and understanding of the problem's graphical representation.

Properties of Logarithms

Understanding the properties of logarithms is essential for solving exponential equations. Logarithms possess several fundamental properties that simplify complex problems. Here’s how these properties helped in solving the equation \( e^{-t} = 0.04 \).Key properties include:

• Logarithm of a Power: The property \( \ln(a^b) = b \cdot \ln(a) \) allows us to bring exponents in front of the logarithm, simplifying expressions. In our original problem, we used this to convert \( \ln(e^{-t}) \) to \( -t \cdot \ln(e) \).
• Logarithmic Identity: Since \( \ln(e) = 1 \), any expression multiplied by \( \ln(e) \) simplifies naturally, as seen when \( -t \cdot \ln(e) \) became \( -t \).

These properties make solving equations involving exponential terms more straightforward by reducing exponents and simplifying the overall equation.

Algebraic Solutions

Solving exponential equations algebraically involves strategic manipulation of the equation using mathematical operations. In our example \( e^{-t} = 0.04 \), the approach includes using logarithms and simplification techniques.Steps typically involved in finding algebraic solutions are:

• Take the logarithm: Apply the natural logarithm to both sides, which helps in bringing down the exponent as a coefficient.
• Simplify the equation: Use the logarithmic properties to simplify. Here, \( \ln(e^{-t}) \) becomes \(-t \cdot \ln(e)\).
• Solve for the variable: Isolate the variable by performing arithmetic operations like division or multiplication on both sides of the equation. We arrived at \( t = -\ln(0.04) \).
• Calculate a numerical solution: Use a calculator to find an approximate numerical value, which in our exercise resulted in \( t \approx 3.22 \).

These algebraic techniques are powerful for systematically arriving at solutions to exponential equations that you might encounter in your studies.