Solving Logarithmic Equations Finding X In Log Base X Of 9 Equals 1/2

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Solve for x in the logarithmic equation log⁡x9=12\log _x 9=\frac{1}{2}. This problem requires a solid understanding of logarithms and their properties. Logarithmic equations can initially appear daunting, but by converting them into their equivalent exponential forms, we can often simplify the process of finding the unknown variable. This article will provide a detailed, step-by-step solution to this equation, ensuring a clear understanding of the underlying principles and techniques involved. We will explore the fundamental relationship between logarithms and exponentials, demonstrating how to convert between these forms to solve for x. Furthermore, we will highlight common pitfalls and strategies for verifying the solution to guarantee its accuracy. Whether you're a student tackling logarithmic equations for the first time or someone looking to refresh your understanding, this guide will equip you with the knowledge and skills to confidently solve similar problems.

Understanding Logarithms

To effectively solve for x in logarithmic equations, it is crucial to first grasp the fundamental concept of logarithms. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, this can be expressed as follows:

If $ \log_b a = c $, then $ b^c = a $

Here, b is the base of the logarithm, a is the argument (the number we are taking the logarithm of), and c is the exponent (the logarithm itself). Understanding this relationship is the cornerstone of solving logarithmic equations. For example, if we have $ \log_2 8 = 3 $, this means that 2 raised to the power of 3 equals 8 (i.e., $ 2^3 = 8 $). This basic principle allows us to convert logarithmic equations into exponential equations, which are often easier to manipulate and solve.

When working with logarithms, several properties and rules can further simplify equations. These include the product rule, quotient rule, power rule, and the change of base formula. However, for the specific problem at hand, the primary focus will be on converting the logarithmic equation into its exponential form. This conversion is the key step in isolating x and finding its value. By mastering this technique, you can tackle a wide array of logarithmic equations with greater confidence and accuracy. Remember, the essence of understanding logarithms lies in recognizing their inverse relationship with exponentiation and applying this knowledge to solve for unknown variables.

Converting the Logarithmic Equation to Exponential Form

In our quest to solve for x, the initial and most critical step is converting the given logarithmic equation, log⁡x9=12\log _x 9=\frac{1}{2}, into its equivalent exponential form. This conversion allows us to transition from the logarithmic domain, which can be less intuitive, to the exponential domain, where the variable x is more readily isolated. As previously established, the fundamental relationship between logarithms and exponentials is defined as:

If $ \log_b a = c $, then $ b^c = a $

Applying this definition to our equation, where b corresponds to x (the base), a corresponds to 9 (the argument), and c corresponds to 12\frac{1}{2} (the logarithm), we can rewrite the equation as:

$ x^{\frac{1}{2}} = 9 $

This transformation is pivotal because it changes the equation from a form where x is part of the logarithmic base to a form where x is raised to a power. In this exponential form, we can more easily manipulate the equation to isolate x and determine its value. The fractional exponent of 12\frac{1}{2} represents the square root, which provides a clear path forward in solving for x. This step highlights the power of understanding the logarithmic-exponential relationship, enabling us to simplify complex equations into more manageable forms. By mastering this conversion technique, you can confidently approach a wide range of logarithmic problems.

Solving the Exponential Equation

Now that we have successfully converted the logarithmic equation log⁡x9=12\log _x 9=\frac{1}{2} into its exponential form, x12=9x^{\frac{1}{2}} = 9, the next crucial step is to solve for x. This involves isolating x on one side of the equation. To achieve this, we need to address the fractional exponent of 12\frac{1}{2}, which, as mentioned earlier, represents the square root. To eliminate the square root, we can raise both sides of the equation to the power of 2. This operation is based on the principle that if two quantities are equal, then raising them to the same power will maintain their equality.

Applying this principle, we raise both sides of the equation x12=9x^{\frac{1}{2}} = 9 to the power of 2:

(x12)2=92(x^{\frac{1}{2}})^2 = 9^2

Using the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}, we simplify the left side of the equation:

x(12imes2)=92x^{(\frac{1}{2} imes 2)} = 9^2

This simplifies to:

x1=92x^1 = 9^2

Therefore, we have:

x=92x = 9^2

Now, we calculate 929^2, which is 9 multiplied by itself:

x=81x = 81

Thus, we have found that x equals 81. This result represents the solution to the exponential equation and, consequently, to the original logarithmic equation. The process of raising both sides of the equation to a power is a common and effective technique for solving exponential equations, and it demonstrates the importance of understanding the properties of exponents in algebraic manipulations.

Verifying the Solution

After solving for x in any equation, it is crucial to verify the solution. This step ensures that the obtained value satisfies the original equation and avoids extraneous solutions, which can sometimes arise in logarithmic and radical equations. In our case, we found that x = 81 for the equation log⁡x9=12\log _x 9=\frac{1}{2}. To verify this solution, we substitute x = 81 back into the original logarithmic equation:

log⁡819=12\log _{81} 9=\frac{1}{2}

This equation asks: To what power must we raise 81 to obtain 9? Since 81 is 929^2, we can rewrite 81 as 929^2 in the logarithm:

log⁡929=12\log _{9^2} 9=\frac{1}{2}

Now, we recall the property of logarithms that states log⁡abc=1blog⁡ac\log _{a^b} c = \frac{1}{b} \log _a c. Applying this property, we have:

12log⁡99=12\frac{1}{2} \log _9 9=\frac{1}{2}

Since log⁡99\log _9 9 is equal to 1 (because 9 raised to the power of 1 is 9), the equation becomes:

12imes1=12\frac{1}{2} imes 1 = \frac{1}{2}

12=12\frac{1}{2} = \frac{1}{2}

This confirms that our solution, x = 81, is indeed correct. The verification process not only validates the solution but also reinforces the understanding of logarithmic properties and their application. By substituting the solution back into the original equation and demonstrating its validity, we can be confident in the accuracy of our result. This step is an essential part of problem-solving and should not be overlooked.

Conclusion

In conclusion, we have successfully solved for x in the logarithmic equation log⁡x9=12\log _x 9=\frac{1}{2}. Our journey involved several key steps, starting with a fundamental understanding of logarithms and their relationship to exponential functions. We began by converting the logarithmic equation into its equivalent exponential form, which allowed us to manipulate the equation more easily. This conversion is a critical technique in solving logarithmic equations, as it transforms the problem into a more familiar algebraic structure.

Next, we addressed the exponential equation x12=9x^{\frac{1}{2}} = 9. To isolate x, we raised both sides of the equation to the power of 2, effectively eliminating the fractional exponent. This step utilized the properties of exponents and the principle of maintaining equality by performing the same operation on both sides of the equation. Through this process, we found that x = 81.

Finally, and perhaps most importantly, we verified our solution. Verification is an essential step in problem-solving, especially in equations involving logarithms or radicals, where extraneous solutions can sometimes occur. By substituting x = 81 back into the original logarithmic equation and confirming that it held true, we ensured the accuracy of our result. This step not only validated our solution but also reinforced our understanding of logarithmic properties.

This comprehensive approach to solving logarithmic equations highlights the importance of understanding fundamental concepts, applying appropriate algebraic techniques, and verifying solutions. By mastering these skills, you can confidently tackle a wide range of mathematical problems involving logarithms and exponents. The solution to the equation log⁡x9=12\log _x 9=\frac{1}{2} is therefore x = 81, a testament to the power of mathematical reasoning and careful problem-solving strategies.