Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$e^{a .5 x}=3^{1-2 x}$$

Step-by-Step Solution

Answer

Exact form: \( \frac{\ln(3)}{0.5 + 2\ln(3)} \); Approximate: 0.407

Step 1: Take the Natural Logarithm

We start by taking the natural logarithm of both sides of the equation \( e^{0.5x} = 3^{1-2x} \). This helps us bring the exponents down, which is crucial for solving the equation. The equation becomes \( 0.5x \ln(e) = (1-2x) \ln(3) \). Since \( \ln(e) = 1 \), we simplify it to \( 0.5x = (1-2x) \ln(3) \).

Step 2: Distribute and Simplify

We distribute \( \ln(3) \) on the right side of the equation: \( 0.5x = \ln(3) - 2x \ln(3) \).

Step 3: Combine Like Terms

Move all terms involving \( x \) to one side of the equation: \( 0.5x + 2x \ln(3) = \ln(3) \). Factor \( x \) from the left side: \( x(0.5 + 2\ln(3)) = \ln(3) \).

Step 4: Solve for \( x \) in Exact Form

Divide both sides by \( 0.5 + 2\ln(3) \) to isolate \( x \): \[ x = \frac{\ln(3)}{0.5 + 2\ln(3)} \]. This is the exact form of the solution.

Step 5: Approximate the Solution

Use a calculator to approximate the solution to the nearest thousandth. Calculate the numerator and denominator separately. \( \ln(3) \approx 1.0986 \). Thus, \( 2\ln(3) \approx 2 \times 1.0986 = 2.1972 \), and the denominator becomes \( 0.5 + 2.1972 = 2.6972 \). Finally, \( x \approx \frac{1.0986}{2.6972} \approx 0.407 \).

Key Concepts

Natural Logarithm

Exponential equations often involve exponents, and the natural logarithm can be a powerful tool to solve these types of equations. The natural logarithm, denoted as \( \ln \), has a special base, \( e \), which is approximately 2.718. It's especially useful because \( \ln(e) = 1 \), simplifying calculations when dealing with equations involving \( e \), like the one in our problem: \( e^{0.5x} = 3^{1-2x} \).Using natural logarithms allows us to transform the exponent into a linear expression, making the equation more manageable. For instance, taking the natural logarithm of both sides of our equation leads to \( 0.5x \ln(e) = (1-2x) \ln(3) \), simplifying due to the property of logarithms that \( \ln(e) = 1 \). So, the equation reduces to \( 0.5x = (1-2x) \ln(3) \), enabling further algebraic manipulation.

Exact Form and Approximation

When solving exponential equations, it's essential to express solutions in both exact and approximate forms, especially for more complex equations.### Exact FormThe exact form provides a precise answer without rounding off any decimal points. It's crucial as it retains the mathematical integrity of the solution. In our solved exercise, the exact solution for \( x \) is \[ x = \frac{\ln(3)}{0.5 + 2\ln(3)} \]. This expression is derived from rearranging the simplified equation and accurately reflects the mathematical relationship.### ApproximationIn many practical scenarios, exact values can be challenging to interpret or utilize directly, especially if they involve irrational numbers. Therefore, we approximate the solution to a manageable decimal form. For our exercise, using a calculator gives us \( \ln(3) \approx 1.0986 \). By substituting into our formula, we find \( x \approx 0.407 \) after computing: \( \frac{1.0986}{2.6972} \). Here, rounding is crucial to ensure accuracy and ease of comprehension, typically to the nearest thousandth.

Solving Exponential Equations

Exponential equations can seem daunting, but a structured approach can simplify the process. The goal is often to isolate the variable in the exponent. Let's dive into the method used to solve our specific equation \( e^{0.5x} = 3^{1-2x} \).### Step-by-Step Solution

• Step 1: Apply the natural logarithm to both sides to break down the exponents, helping to simplify the equation. This transforms it to linear terms.
• Step 2: Distribute any constants, like \( \ln(3) \), across the expanded terms to facilitate combining similar terms.
• Step 3: Gather all variable terms on one side. This often involves factoring the variable out or moving terms around to consolidate them.
• Step 4: Solve for the variable by dividing by any remaining coefficients to find the expression for \( x \) in exact form.
• Step 5: Approximate if necessary, by computing the numerical values of natural logarithms and performing the division.

This systematic approach ensures clarity and precision, essential factors when dealing with exponential relationships.