Problem 10 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(2 e^{12 x}=17\)

Step-by-Step Solution

Answer

The solution is approximately 0.1366.

Step 1: Isolate the Exponential Term

Start by dividing both sides of the equation by 2 to isolate the exponential term. The equation becomes: \[ e^{12x} = \frac{17}{2} \] This simplifies the problem to a standard exponential function form.

Step 2: Apply the Natural Logarithm

Now that we have the exponential term isolated, apply the natural logarithm (\(\ln\)) to both sides of the equation to undo the exponential function: \[ \ln\left( e^{12x} \right) = \ln\left( \frac{17}{2} \right) \] Since the natural logarithm and the exponential function are inverse operations, this simplifies to: \[ 12x = \ln\left( \frac{17}{2} \right) \]

Step 3: Solve for x

To find \(x\), divide both sides of the equation by 12: \[ x = \frac{\ln\left( \frac{17}{2} \right)}{12} \] Use a calculator to find the numerical value of \(\ln\left( \frac{17}{2} \right)\), and then divide by 12 to find \(x\): \[ x \approx \frac{1.6391}{12} \approx 0.1366 \]

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm. It uses the base of Euler's number, which is approximately 2.71828, often denoted as "e." Euler's number is an irrational and transcendental number, frequently seen in mathematical constants and used in natural growth processes. The natural logarithm, noted as "ln," serves the purpose of finding the power to which "e" must be raised to obtain a given number.Why Natural Logarithm?
It is particularly valuable when dealing with exponential equations in the form of \( e^x \), like in our current exercise. By applying the natural logarithm to both sides of an equation, we can "undo" the exponential function, essentially simplifying it:

• \( \ln(e^x) = x \)

This property makes it possible to solve for the variable, helping us transform complex exponential functions into simpler linear equations.Using \( \ln \) in our Exercise
In the exercise, after isolating the exponential term, applying the natural logarithm helped us to transform \( e^{12x} = \frac{17}{2} \) into a more manageable form of \( 12x = \ln\left( \frac{17}{2} \right) \). This is a fundamental step toward finding the solution for \( x \).

Solving for x

Solving for \( x \) is the primary goal in most algebraic equations, including exponential equations. The challenge lies in manipulating the equation to express \( x \) by itself on one side of the equation.Steps to Solve for x in Exponential Equations

• Convert the exponential equation to a logarithmic form. In our case, we use the natural logarithm.
• Apply properties of logarithms to isolate \( x \).
• Simplify the equation until \( x \) stands alone.

This is essential to find the numerical value for \( x \).Detailed Breakdown in Our Exercise
After applying the natural logarithm, we worked with the equation \( 12x = \ln\left( \frac{17}{2} \right) \). Then, it's crucial to solve for \( x \) by dividing both sides by 12:

• \( x = \frac{\ln(\frac{17}{2})}{12} \)

Using a calculator, the natural logarithm \( \ln(\frac{17}{2}) \) results in approximately 1.6391. Thus:

• \( x \approx \frac{1.6391}{12} \approx 0.1366 \)

This solution represents the value of \( x \) required to satisfy the original equation.

Isolation of Variables

Isolation of variables is a technique used to focus on a specific variable within an equation. The goal is to rearrange the equation so that the variable you are interested in is alone on one side of the equation.Why It's Important?
This technique simplifies equations, making them easier to solve. It provides a clear path to find the desired values and is particularly useful in equations involving multiple steps.Applying this Concept in Exponential Equations
In exponential equations like \( 2e^{12x} = 17 \), the first step often involves isolating the exponential term, as shown:

• Divide both sides by 2: \( e^{12x} = \frac{17}{2} \)

Once isolated, the application of a natural logarithm becomes more straightforward, allowing for a logical progression to solve for \( x \).Benefits of Isolation in Our Exercise
Isolation helped us break down the complexity of the problem:

• It reduced the problem to a standard form, facilitating the use of logarithms.
• Simplified the solution path, streamlining our calculations to the final answer.

By isolating \( e^{12x} \), we leveraged both logarithmic principles and fundamental algebraic techniques, successfully solving the exercise.