Solving Logarithmic Equations Finding The Value Of X In $2 Log _x((1-\sqrt{x})/\sqrt{x})=0$

ight)=0$ for $x$

Introduction

In this article, we delve into solving the equation 2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}} ight)=0. This problem combines logarithmic and algebraic concepts, requiring a careful step-by-step approach to find the value(s) of $x$ that satisfy the given condition. Understanding the properties of logarithms and algebraic manipulation is crucial in arriving at the correct solution. We will explore the domain of the function, simplify the equation, and discuss any potential extraneous solutions. This exploration will not only provide the solution but also offer insights into handling similar mathematical problems.

Understanding the Logarithmic Equation

To solve the equation 2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}} ight)=0, we must first understand the domain and properties of logarithms. The logarithmic function $\log_b(a)$ is defined only when $a > 0$, $b > 0$, and $b \neq 1$. Applying these conditions to our equation, we need to ensure that:

1. $x > 0$
2. $x \neq 1$
3. $\frac{1-\sqrt{x}}{\sqrt{x}} > 0$

The third condition, $\frac{1-\sqrt{x}}{\sqrt{x}} > 0$, implies that $1 - \sqrt{x}$ must have the same sign as $\sqrt{x}$. Since $\sqrt{x}$ is always non-negative for $x > 0$, we need $1 - \sqrt{x} > 0$. This inequality can be rewritten as:

$1 > \sqrt{x}$

Squaring both sides (since both sides are positive) gives:

$1 > x$

Combining this with the initial conditions $x > 0$ and $x \neq 1$, we find that the domain of the equation is $0 < x < 1$.

With the domain established, we can now proceed to solve the equation. The equation 2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}} ight)=0 can be simplified by dividing both sides by 2:

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

By the definition of logarithms, $\log_b(a) = 0$ if and only if $a = 1$. Therefore, we have:

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

This equation can be solved algebraically to find the value of $x$.

Solving the Algebraic Equation

Having simplified the logarithmic equation to $\frac{1-\sqrt{x}}{\sqrt{x}} = 1$, we can now focus on solving this algebraic equation. To begin, multiply both sides by $\sqrt{x}$ to eliminate the denominator:

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

Next, add $\sqrt{x}$ to both sides of the equation to isolate the constant term:

$1 = 2\sqrt{x}$

Now, divide both sides by 2:

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

To solve for $x$, square both sides of the equation:

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

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

Thus, we have found a potential solution for $x$: $x = \frac{1}{4}$.

It's essential to verify that this solution lies within the domain we established earlier. The domain was $0 < x < 1$, and since $0 < \frac{1}{4} < 1$, our solution is indeed within the valid domain. Therefore, $x = \frac{1}{4}$ is a valid solution to the given equation.

Verification of the Solution

To ensure the correctness of our solution, we need to substitute $x = \frac{1}{4}$ back into the original equation:

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

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

$2 \log _{\frac{1}{4}}\left(\frac{1-\sqrt{\frac{1}{4}}}{\sqrt{\frac{1}{4}}}\right) = 0$

First, simplify the square root:

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

Now substitute this back into the equation:

$2 \log _{\frac{1}{4}}\left(\frac{1-\frac{1}{2}}{\frac{1}{2}}\right) = 0$

Simplify the fraction inside the logarithm:

$2 \log _{\frac{1}{4}}\left(\frac{\frac{1}{2}}{\frac{1}{2}}\right) = 0$

$2 \log _{\frac{1}{4}}(1) = 0$

Since the logarithm of 1 to any base is 0, we have:

$2 imes 0 = 0$

$0 = 0$

The equation holds true, confirming that $x = \frac{1}{4}$ is indeed a solution to the given equation. This verification step is crucial in mathematical problem-solving to avoid extraneous solutions that may arise during the algebraic manipulations.

Conclusion

In summary, we solved the equation 2 \log _x\left(\frac{1-\sqrt{x}}{\sqrt{x}} ight)=0 by first establishing the domain of the logarithmic function and then simplifying the equation algebraically. The initial conditions required $x > 0$, $x \neq 1$, and $\frac{1-\sqrt{x}}{\sqrt{x}} > 0$. From these conditions, we deduced that the domain is $0 < x < 1$.

By simplifying the equation, we arrived at $\frac{1-\sqrt{x}}{\sqrt{x}} = 1$. Solving this algebraic equation led us to the potential solution $x = \frac{1}{4}$. We verified that this solution falls within the established domain, making it a valid solution.

Finally, we substituted $x = \frac{1}{4}$ back into the original equation to confirm its validity, which proved that $x = \frac{1}{4}$ is indeed the solution. This step-by-step approach highlights the importance of understanding the properties of logarithms, algebraic manipulation, and domain considerations in solving complex equations. This problem serves as a good example of how combining different mathematical concepts can lead to a solution when approached methodically.

Key Takeaways:

• Always consider the domain of logarithmic functions before solving equations.
• Algebraic manipulation should be done carefully to avoid extraneous solutions.
• Verifying the solution is a crucial step in ensuring correctness.

By mastering these techniques, one can confidently tackle similar mathematical challenges and enhance their problem-solving skills.