Solving Logarithmic Equations A Step-by-Step Guide To Solving Log₃(x²) = 7

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In this article, we delve into the fascinating world of logarithmic equations, specifically focusing on solving the equation log₃(x²) = 7. This type of equation involves logarithms, which are the inverse operations of exponentiation. Understanding logarithms is crucial in various fields, including mathematics, physics, computer science, and finance. We will break down the equation step-by-step, explaining the underlying principles and demonstrating the techniques required to find the solution(s). Our exploration will not only provide a solution to this particular problem but also equip you with the knowledge to tackle similar logarithmic equations with confidence. So, let's embark on this mathematical journey and unravel the intricacies of logarithms together.

Understanding Logarithms

Before we dive into solving the equation, let's first establish a solid understanding of what logarithms are. A logarithm answers the question: "To what power must we raise the base to get a certain number?" In the expression logₐ(b) = c, 'a' is the base, 'b' is the argument (the number we want to obtain), and 'c' is the logarithm (the exponent to which we must raise 'a'). This equation is equivalent to the exponential form aᶜ = b. For instance, log₂(8) = 3 because 2³ = 8. Here, 2 is the base, 8 is the argument, and 3 is the logarithm.

The logarithmic equation log₃(x²) = 7 involves a base of 3 and an argument of x². The equation essentially asks: "To what power must we raise 3 to get x²?" The answer, according to the equation, is 7. Therefore, we have the logarithmic equation log₃(x²) = 7. To effectively solve this equation, it's essential to grasp the relationship between logarithmic and exponential forms. This understanding forms the foundation for manipulating and simplifying logarithmic expressions, ultimately leading to the solution. The ability to convert between these forms is a fundamental skill in working with logarithms and will be instrumental in our problem-solving approach.

Converting to Exponential Form

The key to solving the equation log₃(x²) = 7 lies in converting it from logarithmic form to its equivalent exponential form. As we discussed earlier, the logarithmic equation logₐ(b) = c is equivalent to the exponential equation aᶜ = b. Applying this principle to our equation, where the base 'a' is 3, the argument 'b' is x², and the logarithm 'c' is 7, we can rewrite log₃(x²) = 7 as 3⁷ = x². This conversion transforms the equation into a more familiar and manageable form. Now, we have an equation that expresses x² as a power of 3. This form is significantly easier to work with because it eliminates the logarithm, allowing us to directly isolate and solve for x. The conversion from logarithmic to exponential form is a crucial step in solving logarithmic equations, providing a pathway to simplify the problem and find the unknown variable. This technique is widely applicable to various logarithmic equations and serves as a cornerstone in logarithmic problem-solving.

Solving for x²

Now that we have converted the equation to its exponential form, 3⁷ = x², we can proceed to calculate the value of 3⁷. This involves raising 3 to the power of 7, which means multiplying 3 by itself seven times: 3 * 3 * 3 * 3 * 3 * 3 * 3. Performing this calculation, we find that 3⁷ equals 2187. Therefore, our equation now simplifies to x² = 2187. This equation tells us that the square of x is equal to 2187. To find the possible values of x, we need to take the square root of both sides of the equation. Taking the square root is the inverse operation of squaring, and it will isolate x. Remember that when taking the square root of a number, we must consider both the positive and negative roots, as both the positive and negative values, when squared, will yield the same positive result. This is a crucial step to ensure we find all possible solutions to the equation. So, let's move on to finding the square roots of 2187.

Finding the Square Roots

To find the values of x that satisfy the equation x² = 2187, we need to take the square root of both sides. This gives us x = ±√2187. The symbol ± indicates that there are two possible solutions: a positive square root and a negative square root. Now, we need to determine the square root of 2187. The square root of 2187 is not a whole number, so we can express it in its simplest radical form. We can factor 2187 as 3⁷, which can be written as 3⁶ * 3. The square root of 3⁶ is 3³, which equals 27. Therefore, √2187 can be simplified to √(3⁶ * 3) = √(3⁶) * √3 = 3³√3 = 27√3. This means that the square root of 2187 is 27√3. Consequently, the two solutions for x are x = 27√3 and x = -27√3. These are the exact values of x that satisfy the original equation. It's important to express the solutions in their simplest form, which often involves simplifying radicals. Now that we have found the solutions, let's summarize our findings.

Final Solutions

In conclusion, after solving the logarithmic equation log₃(x²) = 7, we have found two solutions for x. These solutions are x = 27√3 and x = -27√3. We arrived at these solutions by first converting the logarithmic equation to its equivalent exponential form, which was 3⁷ = x². We then calculated 3⁷ to be 2187, giving us the equation x² = 2187. Taking the square root of both sides, we obtained x = ±√2187. Finally, we simplified √2187 to 27√3, resulting in the two solutions: x = 27√3 and x = -27√3. These are the values of x that, when squared and used as the argument in the original logarithmic equation, will satisfy the equation. It's important to note that when solving equations involving square roots, we must consider both the positive and negative roots to ensure we capture all possible solutions. This comprehensive step-by-step approach demonstrates the process of solving logarithmic equations, emphasizing the importance of understanding logarithmic properties and the relationship between logarithmic and exponential forms. The solutions we have found are the precise values that make the equation true, completing our mathematical exploration.

In this article, we successfully solved the logarithmic equation log₃(x²) = 7. We began by understanding the fundamentals of logarithms and their relationship to exponential functions. We then converted the logarithmic equation into its equivalent exponential form, which allowed us to isolate x². After calculating the value of 3⁷, we arrived at the equation x² = 2187. Taking the square root of both sides, we found the two solutions for x: x = 27√3 and x = -27√3. This process highlights the importance of understanding logarithmic properties and the ability to convert between logarithmic and exponential forms. The solutions we obtained are the exact values that satisfy the given equation. This exploration not only provides a solution to this specific problem but also equips you with the necessary skills to tackle a variety of logarithmic equations. By mastering these techniques, you can confidently approach logarithmic problems in various mathematical and scientific contexts. Remember, the key to success in mathematics lies in understanding the underlying principles and practicing problem-solving techniques.