Question

The number of integral value of x satisfying the equation $|\log_{\sqrt{3}}x-2|-|\log_3x-2|=2$, is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

1

Answer 2

Here's how to solve the equation $|\log_{\sqrt{3}}x-2|-|\log_3x-2|=2$:

1. Simplify the Logarithms: Recognize that $\log_{\sqrt{3}}x = 2\log_3 x$. This allows us to rewrite the equation as $|2\log_3 x - 2| - |\log_3 x - 2| = 2$.

2. Substitution: Let $y = \log_3 x$. The equation becomes $2|y - 1| - |y - 2| = 2$.

3. Solve the Absolute Value Equation: Consider three cases:

   • Case 1: $y < 1$. Then $2(1 - y) - (2 - y) = 2$, which simplifies to $-y = 2$, so $y = -2$.

   • Case 2: $1 \le y < 2$. Then $2(y - 1) - (2 - y) = 2$, which simplifies to $3y - 4 = 2$, so $y = 2$. However, this solution is not strictly less than 2, so it's not valid in this interval.

   • Case 3: $y \ge 2$. Then $2(y - 1) - (y - 2) = 2$, which simplifies to $y = 2$.

4. Substitute Back: We found $y = -2$ and $y = 2$. Substituting back $y = \log_3 x$:

   • If $\log_3 x = -2$, then $x = 3^{-2} = \frac{1}{9}$.

   • If $\log_3 x = 2$, then $x = 3^2 = 9$.

5. Identify Integral Solutions: Of the two solutions $x = \frac{1}{9}$ and $x = 9$, only $x = 9$ is an integer.

Therefore, there is only 1 integral value of x that satisfies the equation.