Solving E^(3x-3) = 7 A Step-by-Step Guide

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In this article, we will walk through the process of solving exponential equations, specifically focusing on the equation e^(3x-3) = 7. Exponential equations are a fundamental concept in mathematics, and mastering them is crucial for various applications in science, engineering, and finance. We will break down each step, ensuring a clear understanding of how to isolate the variable and obtain an accurate solution, rounded to the nearest thousandth.

Understanding Exponential Equations

Before diving into the specific problem, let's establish a solid understanding of what exponential equations are and the properties that govern them. An exponential equation is an equation in which the variable appears in the exponent. These equations often involve a constant base raised to a power that includes the variable. The general form of an exponential equation is a^f(x) = b, where a and b are constants and f(x) is a function involving x. Solving exponential equations involves isolating the exponential term and then using logarithms to bring the exponent down.

One of the key properties used in solving these equations is the natural logarithm, denoted as ln(x). The natural logarithm is the logarithm to the base e, where e is an irrational number approximately equal to 2.71828. The natural logarithm is particularly useful when dealing with exponential functions involving e because of the property ln(e^x) = x. This property allows us to simplify equations where e is the base. Another important property is the power rule of logarithms, which states that ln(a^b) = b * ln(a). This rule is crucial for bringing the exponent containing the variable down, making it possible to solve for the variable.

To effectively tackle exponential equations, it's essential to grasp the inverse relationship between exponential functions and logarithmic functions. Just as addition and subtraction are inverse operations, and multiplication and division are inverse operations, exponentiation and logarithms undo each other. This inverse relationship is why logarithms are the go-to tool for solving exponential equations. For example, if we have e^x = y, we can take the natural logarithm of both sides to get ln(e^x) = ln(y), which simplifies to x = ln(y). This simple transformation is the backbone of solving more complex exponential equations.

Solving the Equation e^(3x-3) = 7

Now, let's solve the given exponential equation: e^(3x-3) = 7.

Step 1: Apply the Natural Logarithm

The first step in solving this equation is to apply the natural logarithm to both sides. This is a crucial step because it allows us to use the property ln(e^x) = x and bring the exponent down. Applying the natural logarithm to both sides of the equation, we get:

ln(e^(3x-3)) = ln(7)

By applying the natural logarithm, we maintain the equation's balance, as we're performing the same operation on both sides. This is a fundamental principle in solving any equation. The next step involves simplifying the left side using the logarithmic property.

Step 2: Simplify Using Logarithmic Properties

Using the property ln(e^x) = x, we can simplify the left side of the equation. In our case, x is (3x - 3), so we have:

3x - 3 = ln(7)

This simplification is a major step forward because we have removed the exponential part and now have a linear equation in terms of x. The problem has been transformed from solving an exponential equation to solving a linear equation, which is a much simpler task. The natural logarithm has effectively unwrapped the exponent, allowing us to isolate x.

Step 3: Isolate the Variable x

To isolate x, we need to perform algebraic manipulations to get x by itself on one side of the equation. The first step in this process is to add 3 to both sides:

3x - 3 + 3 = ln(7) + 3
3x = ln(7) + 3

By adding 3 to both sides, we've eliminated the constant term on the left side, bringing us closer to isolating x. The equation now reads 3x = ln(7) + 3. The next step involves dividing both sides by the coefficient of x, which is 3.

Step 4: Divide to Solve for x

Now, we divide both sides of the equation by 3:

3x / 3 = (ln(7) + 3) / 3
x = (ln(7) + 3) / 3

This step completes the isolation of x. We now have an expression for x in terms of ln(7) and constants. The exact solution is x = (ln(7) + 3) / 3. However, the problem asks for an answer rounded to the nearest thousandth, so we need to compute the numerical value.

Step 5: Calculate the Numerical Value

To find the numerical value of x, we use a calculator to evaluate ln(7) and then substitute it into the expression. The natural logarithm of 7 is approximately 1.94591.

So, we have:

x = (1.94591 + 3) / 3
x = 4.94591 / 3
x ≈ 1.64864

We have calculated the value of x to several decimal places. The final step is to round this value to the nearest thousandth.

Step 6: Round to the Nearest Thousandth

Rounding 1.64864 to the nearest thousandth (three decimal places), we look at the fourth decimal place, which is 6. Since 6 is greater than or equal to 5, we round up the third decimal place.

Thus, we get:

x ≈ 1.649

The solution to the equation e^(3x-3) = 7, rounded to the nearest thousandth, is approximately 1.649.

Summary of Steps

  1. Apply the Natural Logarithm: Take the natural logarithm of both sides of the equation: ln(e^(3x-3)) = ln(7).
  2. Simplify Using Logarithmic Properties: Use the property ln(e^x) = x to simplify: 3x - 3 = ln(7).
  3. Isolate the Variable x: Add 3 to both sides: 3x = ln(7) + 3.
  4. Divide to Solve for x: Divide both sides by 3: x = (ln(7) + 3) / 3.
  5. Calculate the Numerical Value: Use a calculator to find ln(7) and compute x ≈ 1.64864.
  6. Round to the Nearest Thousandth: Round the value to three decimal places: x ≈ 1.649.

Conclusion

Solving exponential equations involves understanding the properties of logarithms and applying them strategically to isolate the variable. In this article, we meticulously solved the equation e^(3x-3) = 7, demonstrating each step with clear explanations. By applying the natural logarithm, simplifying using logarithmic properties, and performing algebraic manipulations, we arrived at the solution x ≈ 1.649, rounded to the nearest thousandth. This comprehensive guide should enhance your understanding of exponential equations and boost your confidence in tackling similar problems. Mastering these techniques is crucial for success in higher mathematics and various applications in STEM fields. Remember to practice regularly and reinforce your understanding of logarithmic properties to excel in solving exponential equations.