Solving Exponential Equations 1/36 = 6^(x-3) A Step-by-Step Guide
In this article, we will delve into the process of solving an exponential equation. Exponential equations are equations where the variable appears in the exponent. These types of equations are fundamental in various fields, including mathematics, physics, engineering, and finance. The equation we will be tackling today is 1/36 = 6^(x-3). This equation presents a classic example of how to manipulate exponential expressions and apply logarithmic properties to find the unknown variable, x. The aim is to provide a clear, step-by-step solution, ensuring readers understand the underlying principles and can apply them to similar problems. Understanding exponential equations is crucial for anyone studying or working in quantitative fields, as they frequently appear in models of growth, decay, and many other natural phenomena. By the end of this article, you should have a solid grasp of how to approach and solve exponential equations, and feel confident in your ability to tackle related problems.
Before we jump into the solution, it's essential to understand what exponential equations are and the key properties that govern them. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation is a^x = b, where a is the base, x is the exponent (which is the variable we often need to solve for), and b is the result. The base a is usually a positive number not equal to 1, because if a were 1, the equation would simply become 1^x = b, which is not an exponential equation in the traditional sense. Understanding the properties of exponents is crucial for solving these types of equations.
One of the fundamental properties we will use is the ability to rewrite numbers with the same base. This is a critical strategy because if we can express both sides of an equation with the same base, we can then equate the exponents. For example, if we have a^x = a^y, then it follows that x = y. This principle is the cornerstone of solving many exponential equations. Another important property is that a negative exponent indicates the reciprocal of the base raised to the positive exponent. That is, a^(-n) = 1/a^n. This property will be particularly useful in our problem, where we have a fraction on one side of the equation.
Additionally, it's worth mentioning that logarithms are closely related to exponential functions. A logarithm is essentially the inverse operation of exponentiation. If a^x = b, then we can also write this as log_a(b) = x, where log_a represents the logarithm to the base a. While we won't need logarithms for the specific equation we are solving in this article, they are a powerful tool for more complex exponential equations. Grasping these foundational concepts is vital for anyone looking to master exponential equations. These equations are not just theoretical exercises; they have real-world applications in areas such as compound interest calculations, radioactive decay modeling, and population growth studies. By understanding how to manipulate and solve exponential equations, you are equipping yourself with a powerful mathematical tool.
Let's now tackle the equation 1/36 = 6^(x-3) step-by-step. Our goal is to isolate x, the variable in the exponent. The first critical step in solving this exponential equation is to express both sides of the equation with the same base. Recognizing that 36 is a power of 6 is key here. Specifically, 36 is equal to 6 squared, or 6^2. Therefore, we can rewrite the left side of the equation, 1/36, as 1/(6^2).
Now, we can use the property of negative exponents to further rewrite 1/(6^2) as 6^(-2). This property states that a^(-n) = 1/a^n. Applying this, our equation now becomes:
6^(-2) = 6^(x-3)
This is a crucial step because we now have the same base, 6, on both sides of the equation. With the same base, we can equate the exponents. This is based on the principle that if a^m = a^n, then m = n. In our case, this means that we can set the exponents equal to each other:
-2 = x - 3
Now we have a simple linear equation to solve for x. To isolate x, we add 3 to both sides of the equation:
-2 + 3 = x - 3 + 3
This simplifies to:
1 = x
Therefore, the solution to the equation 1/36 = 6^(x-3) is x = 1. This methodical approach, breaking down the problem into manageable steps, is vital for solving exponential equations. Each step, from rewriting the fraction with a negative exponent to equating the exponents and solving the resulting linear equation, is grounded in fundamental mathematical principles. By mastering these steps, you can confidently tackle a wide range of exponential equations.
It's always a good practice to verify the solution to ensure its correctness. This step helps catch any potential errors made during the solving process. To verify our solution, we substitute the value of x we found, which is x = 1, back into the original equation:
1/36 = 6^(x-3)
Substituting x = 1 gives us:
1/36 = 6^(1-3)
Simplify the exponent:
1/36 = 6^(-2)
Now, recall that a^(-n) = 1/a^n. Applying this property, we can rewrite 6^(-2) as 1/(6^2):
1/36 = 1/(6^2)
Since 6^2 = 36, we have:
1/36 = 1/36
This confirms that our solution x = 1 is correct, as it satisfies the original equation. Verification is a crucial step in problem-solving, particularly in mathematics, as it provides assurance that the derived solution is accurate. This process reinforces understanding and builds confidence in one's ability to solve equations correctly. In this case, substituting the solution back into the original equation and simplifying both sides to show equality is a robust method for verification. By adopting this practice, you can minimize errors and strengthen your grasp of mathematical concepts.
While we've solved the equation 1/36 = 6^(x-3) using the method of equating exponents, there are alternative approaches that can be used, particularly if you are familiar with logarithms. Logarithms provide a powerful tool for solving exponential equations where it might not be immediately obvious how to express both sides with the same base. To illustrate, let's consider using logarithms to solve the same equation.
We start with the equation:
1/36 = 6^(x-3)
To apply logarithms, we can take the logarithm of both sides of the equation. It's common to use the natural logarithm (base e, denoted as ln) or the common logarithm (base 10, denoted as log), but any base will work. For simplicity, let's use the natural logarithm:
ln(1/36) = ln(6^(x-3))
Now, we can use the logarithmic property that ln(a^b) = b * ln(a) to simplify the right side of the equation:
ln(1/36) = (x - 3) * ln(6)
Next, we need to isolate (x - 3). To do this, we divide both sides by ln(6):
ln(1/36) / ln(6) = x - 3
Now, we can simplify ln(1/36). Recall that 1/36 = 6^(-2), so ln(1/36) = ln(6^(-2)). Using the logarithmic property again, this becomes:
-2 * ln(6) / ln(6) = x - 3
The ln(6) terms cancel out:
-2 = x - 3
Finally, we add 3 to both sides to solve for x:
-2 + 3 = x
1 = x
We arrive at the same solution, x = 1, confirming our previous result. This alternative approach demonstrates the versatility of logarithms in solving exponential equations. While equating exponents is often the most straightforward method when both sides can be easily expressed with the same base, logarithms offer a powerful alternative when this is not the case. Understanding both methods expands your problem-solving toolkit and allows you to tackle a wider range of exponential equations.
In this article, we've comprehensively explored the solution to the exponential equation 1/36 = 6^(x-3). We began by understanding the fundamentals of exponential equations and the key properties that govern them. We then proceeded with a step-by-step solution, which involved rewriting the equation with a common base, equating the exponents, and solving the resulting linear equation. Our solution yielded x = 1. To ensure the accuracy of our result, we performed a verification step by substituting x = 1 back into the original equation, confirming its validity.
Furthermore, we explored an alternative approach using logarithms. This method showcased the versatility of logarithmic properties in solving exponential equations, providing an additional tool for tackling such problems. The logarithmic approach not only reinforced our solution but also highlighted the interconnectedness of exponential and logarithmic functions.
By mastering the techniques discussed in this article, you are well-equipped to solve a variety of exponential equations. The ability to manipulate exponential expressions and apply logarithmic properties is a valuable skill in many areas of mathematics, science, and engineering. Remember, the key to solving exponential equations lies in understanding the underlying principles, breaking down the problem into manageable steps, and practicing regularly. Whether it's by equating exponents or utilizing logarithms, the methods discussed here provide a solid foundation for your problem-solving journey. With consistent practice and a firm grasp of the fundamentals, you can confidently tackle even more complex exponential equations in the future.
