Question

My question is attached as a word file.

d150b50966c74b40eb1dacc28f55a757.docx

Answer 1

$10^x = x^{50} \\ \\ Log \ 10^x = Log\ x^{50} \\ \\ x\ Log\ 10 = 50 Log\ x \\ \\ x * 1 = 50 * Log\ x \\ \\ Log\ x = \frac{x}{50}$

$Let\ x = 10^y \\ \\ 10^y = 50 * y \\ \\ Log\ 10^y = Log\ 50 * y = Log\ 50 + Log\ y \\ \\ y\ Log\ 10 = Log\ 100/2 + Log\ y \\ \\ y = Log\ 100 - Log\ 2 + Log\ y \\ \\ y - 2 = Log\ y - Log\ 2 \\ \\ put\ y = 2\ then, LHS = RHS \\ \\ So\ \ y = 2 , \ \ \ then \ x = 10^y = 10^2 = 100 \\ \\ verify\ \ \ 10^{100} = 100^{50} = (10^2)^{50} = 10^{100}, it \ verifies.$
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$10^y = 50 * y = 10 * (5y)$

If y is an integer. then,
   as LHS is a multiple of 10, RHS multiple of 10    =>  5 y = 10 or 100 etc.

Then y = 2 or 20 ...
For y = 2,  10² = 10 * 5*2 = 100  =>  it matches.

 so y = 2   so,  x = 100