Question

Product of all the solution of equation $x^{\log_{10}x} = 100 + 2^{\log_2 3} - 3^{\log_3 2})$ is

• A. $\frac{1}{10}$
• B. 1
• C. 10
• D. 100

Answer 1

Answer: (B)

1

Answer 2

First, simplify the equation using the property $a^{\log_a b} = b$:

$2^{\log_2 3} = 3$ and $3^{\log_3 2} = 2$.

So, $100 + 3 - 2 = 101$. The equation becomes:

$x^{\log_{10}x} = 101$

Let $y = \log_{10} x$. Then $\log_{10}(x^{\log_{10} x}) = (\log_{10} x)^2 = y^2$. Applying $\log_{10}$ on both sides:

$y^2 = \log_{10} 101$

This gives two solutions for $y$:

$y = \pm \sqrt{\log_{10} 101}$

Since $x = 10^y$, the solutions are:

$x_1 = 10^{\sqrt{\log_{10}101}}$ and $x_2 = 10^{-\sqrt{\log_{10}101}}$

The product of the solutions is:

$x_1 \cdot x_2 = 10^{\sqrt{\log_{10}101}} \cdot 10^{-\sqrt{\log_{10}101}} = 10^{0} = 1$