Problem 17 - Contemporary Precalculus

Question

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$e^{2 x}=5$$

Step-by-Step Solution

Answer

Answer: The approximate value of x in the equation \(e^{2x} = 5\) is 0.8047.

Step 1: Take the natural logarithm of both sides

Given equation: $$e^{2 x}=5$$ To take the natural logarithm of both sides, we have: $$\ln(e^{2 x}) = \ln(5)$$

Step 2: Solve the equation for x

Using the property of logarithms, we can write the left-hand side as: $$2x \cdot \ln(e) = \ln(5)$$ Since the natural logarithm of e is 1 (\(\ln(e) = 1\)), the equation becomes: $$2x = \ln(5)$$ Now, to solve for x, divide both sides by 2: $$x = \frac{\ln(5)}{2}$$

Step 3: Write the solution in terms of natural logarithms

The solution in terms of natural logarithms is: $$x = \frac{\ln(5)}{2}$$

Step 4: Use a calculator to find the approximation

Using a calculator, we find an approximation for the solution: $$x \approx \frac{\ln(5)}{2} \approx 0.8047$$ So, the approximate solution is: $$x \approx 0.8047$$

Key Concepts

Natural Logarithms

Natural logarithms, often denoted as \( \ln \), are a type of logarithm with a base of the mathematical constant \( e \), which is approximately equal to 2.71828. In the context of solving equations like \( e^{2x} = 5 \), natural logarithms are used to unravel the exponent. By taking the natural logarithm of both sides of an equation, we can simplify and bring down the exponent, so it becomes a part of a simpler linear equation to solve.

• The formula for a natural logarithm is \( \ln(e^y) = y \), which simplifies solving exponential equations.
• Remember, the natural logarithm of \( e \), \( \ln(e) \), is always 1, simplifying expressions where \( e \) is the base.

In the problem, applying \( \ln \) to both sides is an efficient method to isolate \( x \), making the equation manageable.

Properties of Logarithms

Logarithms possess several properties that make solving equations more approachable. Understanding these properties allows us to manipulate equations effectively.

• Property 1: \( \ln(ab) = \ln(a) + \ln(b) \) - This property is useful for breaking down multiplication inside a logarithm.
• Property 2: \( \ln(a/b) = \ln(a) - \ln(b) \) - This is handy for simplifying divisions inside the logarithm.
• Property 3: \( \ln(a^b) = b \cdot \ln(a) \) - This property was utilized in transforming the original equation by moving the exponent down as a multiplier.

In the exercise provided, utilizing \( \ln(e^{2x}) = 2x \cdot \ln(e) \) made it possible to easily solve for \( x \). Recognizing and using these logarithmic properties is key in simplifying exponential equations.

Using Calculators for Approximations

Once we have expressed the solution in terms of natural logarithms, we often need a decimal approximation for practical purposes. This is where calculators become essential.
Using a calculator, you can input the logarithmic expression directly to obtain an approximate decimal value.

• For example, to find an approximation for \( x = \frac{\ln(5)}{2} \), one would enter the expression into a scientific calculator using the \( \ln \) function.
• It's crucial to ensure your calculator is set to handle natural logarithms correctly, as some calculators default to base 10 logarithms (common logarithms).

In practice, this aspect of using calculators aids in transforming the theoretical solution into a tangible numerical answer, like approximating \( x \approx 0.8047 \) for our given problem.