Solving Logarithmic Equations Find X Value In 2 Logx((1-√x)/√x)=0

Introduction

In this comprehensive exploration, we will delve into the intricacies of solving a logarithmic equation. Specifically, we aim to determine the value of x that satisfies the equation $2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right)=0$. Logarithmic equations, with their unique properties and rules, often present intriguing challenges. Understanding the fundamental principles of logarithms and their manipulation is crucial for effectively tackling such problems. This article will not only provide a step-by-step solution but also offer a detailed explanation of the underlying concepts and techniques involved. By the end of this discussion, you will gain a deeper understanding of logarithmic equations and how to solve them systematically.

The equation we are tasked with solving involves several key components: a logarithm with a base x, a fractional argument containing square roots, and a constant multiplier. To unravel this equation, we will employ a combination of algebraic manipulation and logarithmic identities. Our approach will involve simplifying the equation, isolating the logarithmic term, and then converting the equation into an exponential form. Additionally, we will need to consider the domain of the logarithmic function to ensure that our solution is valid. This process will require careful attention to detail and a solid grasp of logarithmic principles. As we progress, we will highlight the key steps and reasoning behind each transformation, making the solution process transparent and easy to follow. By working through this problem, we aim to enhance your problem-solving skills and deepen your understanding of logarithms.

Understanding Logarithmic Equations

Before diving into the solution, let's establish a clear understanding of logarithmic equations. A logarithmic equation is an equation in which a logarithm appears. The logarithm is the inverse operation to exponentiation. For example, if $b^y = x$, then we can write this in logarithmic form as $\log_b(x) = y$, where b is the base, x is the argument, and y is the exponent. The base b must be a positive real number not equal to 1, and the argument x must be a positive real number.

The logarithmic function is defined as $y = \log_b(x)$, where x > 0 and b > 0, b ≠ 1. The domain of the logarithmic function is the set of all positive real numbers, and the range is the set of all real numbers. Understanding these constraints is crucial when solving logarithmic equations, as they help us identify any extraneous solutions that might arise during the solution process. For instance, if we obtain a solution that makes the argument of the logarithm negative or zero, that solution must be discarded.

One of the key properties of logarithms that we will use in solving the equation is the power rule, which states that $\log_b(x^k) = k \log_b(x)$. This rule allows us to move exponents from the argument of the logarithm to the front as a coefficient, and vice versa. Another important property is the change of base formula, which states that $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$, where c is any positive real number not equal to 1. This formula is particularly useful when dealing with logarithms with different bases, as it allows us to convert them to a common base. However, in this specific problem, we will primarily use the basic definition of logarithms and algebraic manipulation to find the solution.

Step-by-Step Solution

1. Initial Equation: Start with the given equation:

$2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right)=0$

This is the equation we need to solve for x. The first step in solving any equation is to simplify it as much as possible. In this case, we can start by dividing both sides of the equation by 2. This will help isolate the logarithmic term and make the subsequent steps easier to manage. Dividing both sides by a constant is a fundamental algebraic operation that preserves the equality of the equation, allowing us to proceed with further simplification.

2. Divide by 2: Divide both sides of the equation by 2:

$\log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right)=0$

Now, we have a simplified equation with the logarithm isolated on one side. The next step is to convert the logarithmic equation into its equivalent exponential form. This is a crucial step in solving logarithmic equations, as it allows us to eliminate the logarithm and work with a more familiar algebraic expression. The definition of a logarithm states that if $\log_b(x) = y$, then $b^y = x$. We will apply this definition to our equation to transform it into an exponential form.

3. Convert to Exponential Form: Use the definition of logarithm to convert the equation to exponential form:

$x^0 = \frac{1-\sqrt{x}}{\sqrt{x}}$

Here, we have used the fact that $\log_b(x) = y$ is equivalent to $b^y = x$. In our case, the base is x, the exponent is 0, and the argument is $\frac{1-\sqrt{x}}{\sqrt{x}}$. This transformation is a key step in solving the equation, as it allows us to eliminate the logarithm and work with a more straightforward algebraic expression. Now, we can simplify the left side of the equation, knowing that any non-zero number raised to the power of 0 is 1.

4. Simplify: Simplify the equation, noting that $x^0 = 1$ (for $x \neq 0$):

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

At this point, we have an algebraic equation involving a square root. To solve for x, we need to isolate the square root term and then square both sides of the equation. This will eliminate the square root and allow us to solve for x. However, before we proceed, it is important to consider the domain of the original equation. The argument of the logarithm, $\frac{1-\sqrt{x}}{\sqrt{x}}$, must be positive, and the base x must be positive and not equal to 1. These conditions will help us determine the valid range of values for x and identify any extraneous solutions.

5. Multiply by $\sqrt{x}$: Multiply both sides by $\sqrt{x}$ to eliminate the denominator:

$\sqrt{x} = 1 - \sqrt{x}$

Multiplying both sides of the equation by $\sqrt{x}$ is a standard algebraic technique to eliminate the denominator. This simplifies the equation and allows us to isolate the square root term more easily. Now, we have an equation where the square root of x appears on both sides. The next step is to collect the square root terms on one side of the equation to further simplify and solve for x.

6. Isolate $\sqrt{x}$: Add $\sqrt{x}$ to both sides:

$2\sqrt{x} = 1$

Adding $\sqrt{x}$ to both sides of the equation is a simple algebraic manipulation that helps isolate the square root term. This brings us closer to solving for x. Now, we have an equation where the square root of x is multiplied by a constant. To further isolate the square root, we will divide both sides of the equation by the constant.

7. Divide by 2: Divide both sides by 2:

$\sqrt{x} = \frac{1}{2}$

Dividing both sides by 2 isolates the square root term completely. Now, we have a simple equation stating that the square root of x is equal to 1/2. To solve for x, we need to eliminate the square root. This can be done by squaring both sides of the equation.

8. Square Both Sides: Square both sides of the equation to solve for $x$:

$x = \left(\frac{1}{2}\right)^2$

Squaring both sides of the equation is the final step in solving for x. This eliminates the square root and gives us the value of x directly. Now, we just need to simplify the right side of the equation to obtain the solution.

9. Final Answer: Simplify the expression:

$x = \frac{1}{4}$

Therefore, the value of x that satisfies the given equation is 1/4. However, we need to check if this solution is valid by ensuring that it satisfies the domain conditions of the original logarithmic equation. The base x must be positive and not equal to 1, and the argument of the logarithm, $\frac{1-\sqrt{x}}{\sqrt{x}}$, must be positive. Let's verify these conditions.

10. Verify the Solution: Check if $x = \frac{1}{4}$ is a valid solution:

• Base Condition: $x = \frac{1}{4}$ is positive and not equal to 1, so it satisfies the base condition.

• Argument Condition: We need to check if $\frac{1-\sqrt{x}}{\sqrt{x}} > 0$ when $x = \frac{1}{4}$.

  $\frac{1-\sqrt{\frac{1}{4}}}{\sqrt{\frac{1}{4}}} = \frac{1-\frac{1}{2}}{\frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$

  Since 1 > 0, the argument condition is also satisfied.

Final Answer

Thus, the value of x that satisfies the equation $2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right)=0$ is $\frac{1}{4}$. This solution is valid because it meets all the conditions imposed by the logarithmic function, namely that the base is positive and not equal to 1, and the argument is positive. By carefully following the steps outlined above, we have successfully solved the logarithmic equation and determined the value of x.

Conclusion

In conclusion, solving logarithmic equations requires a solid understanding of logarithmic properties, algebraic manipulation, and domain considerations. By systematically applying these principles, we can effectively solve complex equations and arrive at accurate solutions. In this article, we successfully determined that the value of x that satisfies the equation $2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right)=0$ is $\frac{1}{4}$. This was achieved through a step-by-step process that involved simplifying the equation, converting it to exponential form, isolating the variable, and verifying the solution against the domain restrictions.

The techniques and concepts discussed in this article are applicable to a wide range of logarithmic equations. By mastering these skills, you can confidently tackle similar problems and enhance your problem-solving abilities in mathematics. Remember to always check your solutions against the domain of the logarithmic function to ensure their validity. The process of solving logarithmic equations is not just about finding the answer; it's about understanding the underlying principles and developing a systematic approach to problem-solving. This understanding will serve you well in more advanced mathematical studies and real-world applications where logarithms play a crucial role.

This exploration has provided a detailed walkthrough of solving a specific logarithmic equation, highlighting the key steps and reasoning involved. We hope that this discussion has been informative and has deepened your understanding of logarithms and their applications. Keep practicing and exploring different types of logarithmic equations to further solidify your skills and knowledge.