Solving 0.3 * E^(3x) = 27 A Step-by-Step Guide

by THE IDEN 47 views

In this article, we will tackle the exponential equation 0.3 * e^(3x) = 27. Our primary goal is to solve for x and express the solution in terms of a natural logarithm (base-e). Furthermore, we will approximate the value of x to the nearest thousandth, providing a practical understanding of the solution. This problem falls under the domain of exponential equations, which are fundamental in various fields such as mathematics, physics, engineering, and finance. Understanding how to solve these equations is crucial for modeling and analyzing real-world phenomena that exhibit exponential growth or decay.

Step-by-Step Solution

1. Isolate the Exponential Term

The initial step in solving the equation 0.3 * e^(3x) = 27 is to isolate the exponential term, e^(3x). To achieve this, we divide both sides of the equation by 0.3:

0.  3 * e^(3x) / 0.3 = 27 / 0.3

This simplifies to:

e^(3x) = 90

Isolating the exponential term allows us to directly address the exponent, which contains our variable x. This step is crucial as it sets the stage for using logarithms to undo the exponential operation.

2. Apply the Natural Logarithm

To eliminate the exponential function with base e, we apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the logarithm to the base e, and it is the inverse function of e^x. Applying the natural logarithm gives us:

ln(e^(3x)) = ln(90)

Using the property of logarithms that ln(e^a) = a, we simplify the left side of the equation:

3x = ln(90)

This transformation is a key step because it brings the variable x down from the exponent, making it easier to solve for.

3. Solve for x

Now that we have 3x = ln(90), we can solve for x by dividing both sides of the equation by 3:

x = ln(90) / 3

This gives us the exact solution for x in terms of the natural logarithm. This form is preferred in many mathematical contexts as it provides an accurate representation of the solution without rounding errors. The expression ln(90) / 3 represents the precise value of x that satisfies the original equation.

4. Approximate the Value of x

To obtain a numerical approximation for x, we use a calculator to evaluate ln(90) / 3. The natural logarithm of 90 is approximately 4.49981. Dividing this by 3, we get:

x ≈ 4.49981 / 3 ≈ 1.49994

Rounding this to the nearest thousandth, we have:

x ≈ 1.500

Therefore, the approximate value of x that satisfies the equation 0.3 * e^(3x) = 27 is 1.500. This approximation provides a practical understanding of the magnitude of x.

Expressing the Solution

Exact Solution

The exact solution for x is expressed in terms of the natural logarithm:

x = ln(90) / 3

This form maintains the precision of the solution and is often preferred in theoretical or symbolic computations.

Approximate Solution

The approximate value of x, rounded to the nearest thousandth, is:

x ≈ 1.500

This numerical approximation is useful for practical applications where a decimal value is required.

Verification

To verify our solution, we can substitute the approximate value of x back into the original equation:

0.  3 * e^(3 * 1.500) = 0.3 * e^(4.5)

Using a calculator, we find that:

0.  3 * e^(4.5) ≈ 0.3 * 90.01713 ≈ 27.005

This result is very close to 27, which confirms that our approximate solution x ≈ 1.500 is accurate to the nearest thousandth. The slight discrepancy is due to rounding errors in the approximation.

Alternative Methods

While the method described above is the most straightforward way to solve this equation, there are alternative approaches that can be used.

Graphical Method

One alternative method is to solve the equation graphically. We can graph the functions y = 0.3 * e^(3x) and y = 27 and find the point of intersection. The x-coordinate of the intersection point will be the solution to the equation. This method is particularly useful for visualizing the solution and understanding the behavior of the functions involved.

Numerical Methods

Another approach is to use numerical methods, such as the Newton-Raphson method, to approximate the solution. These methods involve iterative calculations that converge to the root of the equation. Numerical methods are especially useful for solving equations that do not have closed-form solutions.

Common Mistakes

When solving exponential equations, it is essential to avoid common mistakes that can lead to incorrect solutions. Some of the most common errors include:

  1. Incorrectly applying the order of operations: Ensure that operations are performed in the correct order (PEMDAS/BODMAS). For instance, the exponential term must be evaluated before multiplication.
  2. Misunderstanding logarithm properties: Logarithmic properties, such as ln(e^a) = a, must be applied correctly. Incorrect application of these properties can lead to significant errors.
  3. Rounding errors: Rounding intermediate results can introduce inaccuracies in the final answer. It is best to keep intermediate values in their exact form until the final calculation.
  4. Forgetting to isolate the exponential term: Before applying logarithms, the exponential term must be isolated. Failing to do so can complicate the solution process.

Real-World Applications

Exponential equations have numerous real-world applications in various fields. Some notable examples include:

  1. Population growth: Exponential functions are used to model population growth, where the rate of increase is proportional to the current population.
  2. Radioactive decay: The decay of radioactive substances follows an exponential decay model, where the amount of substance decreases exponentially over time.
  3. Compound interest: The growth of money in an account with compound interest can be modeled using exponential functions.
  4. Chemical reactions: The rates of certain chemical reactions can be described using exponential equations.
  5. Spread of diseases: The spread of infectious diseases can be modeled using exponential functions, especially in the early stages of an outbreak.

Conclusion

In this article, we have successfully solved the equation 0.3 * e^(3x) = 27 for x. We found the exact solution in terms of the natural logarithm, x = ln(90) / 3, and approximated the value to the nearest thousandth, x ≈ 1.500. We also verified our solution and discussed alternative methods and common mistakes to avoid. Understanding how to solve exponential equations is crucial for various applications in mathematics, science, and engineering. The step-by-step approach outlined in this article provides a solid foundation for tackling similar problems.

By isolating the exponential term, applying the natural logarithm, and solving for x, we can accurately determine the value that satisfies the equation. Whether dealing with population growth, radioactive decay, or financial investments, the ability to solve exponential equations is an invaluable skill. Remember to verify your solutions and be mindful of potential errors to ensure the accuracy of your results.