Problem 16 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$e^{4 t}=200$$

Step-by-Step Solution

Answer

To solve the exponential equation \(e^{4t} = 200\) algebraically, first take the natural logarithm of both sides: \(\ln(e^{4t}) = \ln(200)\). Simplify using logarithmic properties and isolate t: \(4t = \ln(200)\) which gives \(t = \frac{\ln(200)}{4} \approx 1.279\). To check the solution, use a graphing calculator to plot \(y = e^{4t}\) and verify that the value of y at t = 1.279 is very close to 200, confirming the algebraic solution's correctness.

Step 1: Take the natural logarithm of both sides

To begin solving the expression algebraically, we take the natural logarithm of both sides. This will help us isolate t and find its value. \[\ln(e^{4t}) = \ln(200)\]

Step 2: Use properties of logarithms to simplify the expression

Now, we will use the properties of logarithms to simplify the left side of the equation. The property we will use here is: \(\ln(a^b) = b\cdot\ln(a)\). Applying this property, we get: \[4t\cdot\ln(e) = \ln(200)\] Since the natural logarithm of e is 1 (\(\ln(e) = 1\)), the left side simplifies to: \[4t = \ln(200)\]

Step 3: Isolate the variable t

To obtain the value of t, we will isolate it by dividing both sides of the equation by 4: \[t = \frac{\ln(200)}{4}\]

Step 4: Compute the result

Now we can compute the value of t: \[t = \frac{\ln(200)}{4} \approx 1.279\]

Step 5: Check the solution using a graphing calculator

To verify our solution using a graphing calculator, we will plot the function \(y = e^{4t}\) and observe if the value of y at t = 1.279 is indeed 200. Plug t = 1.279 into the function: \[y = e^{4(1.279)}\] Approximately, \(y \approx 200\) The graph of the function confirms that the value of y at t = 1.279 is very close to 200. Therefore, our algebraic solution is correct.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). This particular type of logarithm is used frequently in calculus and various scientific applications because of its unique properties. It offers a way to interpret logarithms in continuous growth processes, such as population growth, radioactive decay, and financial calculations.

• Using \( \ln \) allows us to express powers and exponential relationships in a linear form, making them easier to solve.
• The notation \( \ln(e) = 1 \) arises from the fact that \( e^1 = e \).
• Natural logarithms transform multiplicative processes into additive ones, which simplifies the process of dealing with powers and exponents.

In our solved example, we applied the natural logarithm to both sides of the equation to facilitate isolating the variable \( t \). This step is crucial for transforming the exponential equation into a form suitable for straightforward algebraic manipulation.

Properties of Logarithms

Logarithms have several useful properties that can simplify the manipulation of equations, especially when dealing with exponentials. These properties turn potentially complex arithmetic operations into simpler algebraic ones.

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

These properties were essential in our original problem when simplifying \( \ln(e^{4t}) \) to \( 4t \cdot \ln(e) \) using the power property. This simplification helped isolate the variable \( t \) on one side of the equation.

Graphing Calculator

A graphing calculator is a powerful tool that can help verify solutions found algebraically. It allows students to visualize equations and solutions, confirming their findings graphically. Here’s how you can use a graphing calculator to check an exponential equation like \( e^{4t} = 200 \):

• First, plot the function \( y = e^{4t} \).
• Identify the horizontal line corresponding to \( y = 200 \).
• Find the x-coordinate where the plotted function meets \( y = 200 \).
• This intersection point should match the value obtained algebraically (\( t \approx 1.279 \) in this case).

Using a graphing calculator provides a visual confirmation that is often intuitive. It shows whether the value of the function at a specific point corresponds to the expected result, confirming the accuracy of the algebraically derived solution.

Solving Algebraically

Solving an equation algebraically involves manipulating it using mathematical operations to find the unknown variable. The goal is to isolate the variable on one side, making it easy to solve. Here's a simple framework for solving exponential equations like \( e^{4t} = 200 \):

• Take the logarithm of both sides to simplify the powers.
• Use properties of logarithms to handle exponents and simplify expressions.
• Rearrange the equation to isolate the variable of interest.
• Compute the numeric solution, providing an exact or approximate value.

In our problem, taking \( \ln \) of both sides was the first and most crucial step. This step helped transition from an exponential form (\( e^{4t} \)) to a linear form (\( 4t = \ln(200) \)). Once in linear form, isolating \( t \) required basic algebraic division. Finally, using a calculator gave a numerical solution, verifying the steps taken.