Solving Exponential Equations: Find X In 3^(4x) = 27^(x-3)

In this article, we delve into the realm of exponential equations, specifically focusing on solving the equation 3^(4x) = 27^(x-3) for the value of x. Exponential equations, characterized by variables appearing in exponents, present unique challenges and opportunities for mathematical exploration. Understanding how to solve these equations is crucial for various applications in fields such as physics, engineering, and finance. This exploration not only enhances our algebraic skills but also provides a foundation for tackling more complex mathematical problems.

Before diving into the solution, it’s essential to understand the nature of exponential equations. An exponential equation is one in which the variable appears in the exponent. The key to solving such equations lies in manipulating them to have the same base on both sides. Once the bases are the same, we can equate the exponents and solve for the variable. This method stems from the fundamental property of exponential functions, which states that if a^m = a^n, then m = n, provided that a is a positive real number not equal to 1. This property allows us to transition from dealing with exponential expressions to algebraic equations, which are often simpler to solve.

Consider the equation at hand: 3^(4x) = 27^(x-3). The first step in solving this equation involves recognizing that both 3 and 27 can be expressed as powers of the same base. This is a crucial observation, as it sets the stage for simplifying the equation. In this case, 27 can be written as 3^3. Recognizing such relationships between numbers is a fundamental skill in solving exponential equations. By expressing both sides of the equation in terms of the same base, we pave the way for equating the exponents and finding the value of x. The subsequent steps involve algebraic manipulation, which requires a solid understanding of algebraic principles.

The core strategy for solving this exponential equation is to express both sides with the same base. We observe that 27 can be written as 3 cubed, or 3^3. Rewriting the equation using this fact, we have:

3^(4x) = (3³⁾(x-3)

This transformation is a crucial step because it allows us to apply the power of a power rule, which states that (a^m)n = a^(m*n). By applying this rule, we can simplify the right side of the equation and bring the exponents into a form that can be easily compared. This step demonstrates the importance of recognizing and utilizing fundamental exponent rules in solving exponential equations. The ability to manipulate exponential expressions using these rules is a key skill in algebra and is frequently used in various mathematical contexts. In this particular case, the power of a power rule allows us to eliminate the outer exponent on the right side of the equation, making it much easier to equate the exponents and solve for x.

Applying the power of a power rule, we multiply the exponents on the right side:

3^(4x) = 3^(3(x-3))

This simplification is a direct application of the exponent rule and is a common technique in handling exponential equations. It highlights the importance of mastering these rules for efficient problem-solving. The resulting equation now has the same base on both sides, which is a critical milestone in the solution process. With the bases aligned, we are in a position to equate the exponents, transforming the exponential equation into a linear equation. This transition is a significant step because linear equations are generally easier to solve than exponential equations. The ability to make this transformation is a key reason why manipulating exponential equations to have the same base is such a powerful strategy.

Since the bases are now the same, we can equate the exponents:

4x = 3(x-3)

This step is a direct consequence of the fundamental property of exponential functions. When exponential expressions with the same base are equal, their exponents must also be equal. This principle is the cornerstone of solving exponential equations by this method. By equating the exponents, we transform the problem from one involving exponential expressions to a simpler algebraic equation. This transformation is a key strategy in mathematics: reducing a complex problem to a simpler, more manageable one. The resulting equation, 4x = 3(x-3), is a linear equation that can be solved using standard algebraic techniques, such as distribution and rearrangement of terms.

Now, we solve the resulting linear equation:

4x = 3x - 9

Subtracting 3x from both sides gives:

x = -9

This solution is obtained through straightforward algebraic manipulation, demonstrating the power of transforming exponential equations into linear equations. The process involves isolating the variable x by performing the same operations on both sides of the equation, a fundamental principle in algebra. The result, x = -9, is the value that satisfies the original exponential equation. This value can be checked by substituting it back into the original equation to ensure that both sides are equal. The ability to solve linear equations is a crucial skill in mathematics, and it is often a necessary step in solving more complex problems, including exponential equations.

To verify our solution, we substitute x = -9 back into the original equation:

3^(4*(-9)) = 27^(-9-3)

3^(-36) = 27^(-12)

Since 27 = 3^3, we can rewrite the right side as:

3^(-36) = (3³⁾(-12)

3^(-36) = 3^(-36)

This confirms that our solution is correct. Verification is a crucial step in problem-solving, as it ensures that the solution obtained satisfies the original conditions of the problem. In this case, substituting x = -9 back into the original equation and simplifying both sides demonstrates that the equation holds true. This not only validates the solution but also reinforces the understanding of the mathematical principles and steps involved in the solution process. Verification helps to catch any potential errors and builds confidence in the accuracy of the final answer.

The value of x that satisfies the equation 3^(4x) = 27^(x-3) is -9. This problem illustrates a common strategy for solving exponential equations: rewriting the equation so that both sides have the same base, then equating the exponents. This method is widely applicable and is a cornerstone of solving exponential equations. By mastering this technique, one can tackle a wide range of problems involving exponential functions. The process involves a combination of recognizing numerical relationships, applying exponent rules, and utilizing algebraic manipulation to isolate the variable. This approach not only provides the solution to the specific problem but also enhances overall mathematical problem-solving skills.

Solving Exponential Equations Find x in 3^(4x) = 27^(x-3)