Solving For X In Exponential Equations A Comprehensive Guide

In mathematics, solving for variables in equations is a fundamental skill. Exponential equations, where the variable appears in the exponent, present a unique challenge. This article delves into solving such an equation step-by-step, providing a clear and thorough explanation suitable for students and enthusiasts alike. We will address the equation

${ \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{3x} \times \left(\frac{3}{5}\right)^{2x-1} }$

and find the value of x. This exploration will not only solve the problem but also illuminate the underlying principles of exponential functions and equation manipulation.

Understanding Exponential Equations

When dealing with exponential equations, a solid grasp of exponential properties is crucial. These properties allow us to simplify complex expressions and isolate the variable. An exponential equation is one in which the variable appears in the exponent. The key to solving these equations often involves manipulating them so that terms with the same base can be easily compared. This is because if we have an equation of the form a**m = a**n, then it follows that m = n. This fundamental principle allows us to move from exponential comparisons to algebraic equations, which are often simpler to solve.

To effectively solve exponential equations, it’s essential to understand and apply the rules of exponents. These rules include the product of powers rule (a**m * an* = a(m+n)*), the quotient of powers rule (*am* / *an* = a(m-n)), the power of a power rule (( a**m )n = a(mn)), and the negative exponent rule (a*-n = 1 / a**n). Recognizing and applying these rules is pivotal in simplifying and solving exponential equations. For instance, the product of powers rule is particularly useful when multiplying exponential terms with the same base, as seen in the given equation. Understanding these properties is not just about memorization; it’s about recognizing how they can be strategically applied to transform complex equations into simpler, solvable forms. This mastery of exponential rules builds a strong foundation for tackling more advanced mathematical problems involving exponential functions.

Detailed Solution

Step 1: Apply the Product of Powers Rule

The first key step in solving the equation is to simplify the right-hand side. We utilize the product of powers rule, which states that when multiplying exponential expressions with the same base, we add the exponents. In our case, the base is (3/5), and the exponents are 3x and (2x - 1). Applying this rule, we rewrite the equation as:

${ \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{3x + (2x - 1)} }$

This step is crucial as it consolidates the two exponential terms on the right-hand side into a single term. By adding the exponents, we simplify the equation, making it easier to compare both sides. The product of powers rule is a fundamental property in algebra, and its application here highlights its importance in simplifying exponential expressions. Recognizing this opportunity to combine terms is key to progressing towards a solution.

Step 2: Simplify the Exponent

Having applied the product of powers rule, the next step is to simplify the exponent on the right-hand side of the equation. This involves combining like terms in the expression 3x + (2x - 1). By adding the x terms and keeping the constant term, we simplify the exponent as follows:

${ 3x + (2x - 1) = 3x + 2x - 1 = 5x - 1 }$

Thus, the equation now looks like this:

${ \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{5x - 1} }$

This simplification is a critical step because it reduces the complexity of the equation, making it more manageable. By combining like terms, we clarify the relationship between the exponents on both sides of the equation. This process of simplification is a common strategy in algebra, allowing us to break down complex expressions into more understandable components. The simplified exponent now presents a clearer path towards solving for x.

Step 3: Equate the Exponents

Now that we have the same base on both sides of the equation, we can equate the exponents. This is a fundamental principle in solving exponential equations: if a**m = a**n, then m = n. Applying this principle to our equation:

${ \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{5x - 1} }$

we can equate the exponents:

${ -6 = 5x - 1 }$

This step is crucial as it transforms the exponential equation into a simple linear equation. By equating the exponents, we eliminate the exponential part of the equation, leaving us with a straightforward algebraic problem to solve. This transition from an exponential equation to a linear equation is a key technique in solving these types of problems. The resulting linear equation is much easier to handle and allows us to isolate x using basic algebraic manipulations.

Step 4: Solve for x

With the equation now simplified to a linear form, we can proceed to solve for x. The equation we have is:

${ -6 = 5x - 1 }$

To isolate x, we first add 1 to both sides of the equation:

${ -6 + 1 = 5x - 1 + 1}$ ${ -5 = 5x }$

Next, we divide both sides by 5:

${ \frac{-5}{5} = \frac{5x}{5} }$ ${ -1 = x }$

Thus, the solution is x = -1. This step-by-step algebraic manipulation demonstrates the process of isolating the variable, a fundamental skill in solving equations. By performing these operations carefully, we arrive at the value of x that satisfies the original exponential equation. This final solution represents the culmination of the simplification and equation-solving techniques applied throughout the process.

Conclusion

In summary, we have successfully solved the exponential equation:

${ \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{3x} \times \left(\frac{3}{5}\right)^{2x-1} }$

and found that x = -1. The process involved applying the product of powers rule, simplifying exponents, equating exponents, and solving the resulting linear equation. This comprehensive walkthrough illustrates the importance of understanding exponential properties and algebraic manipulation in solving mathematical problems.

Solving exponential equations requires a clear understanding of exponential rules and algebraic techniques. This example highlights how to systematically approach such problems, breaking them down into manageable steps. The application of the product of powers rule, the simplification of exponents, and the transformation into a linear equation are key strategies in this process. The final solution, x = -1, is a testament to the power of these methods. By mastering these techniques, students and enthusiasts can confidently tackle a wide range of exponential equations, building a strong foundation in mathematics. Understanding these concepts not only aids in solving equations but also enhances problem-solving skills applicable in various fields.

Final Answer

Therefore, the value of x that satisfies the equation is -1.