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Question

If $open parentheses x plus 1 close parentheses to the power of log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses end exponent equals 100 open parentheses x plus 1 close parentheses comma$ then

All the roots are positive real numbers.
All the roots lie in the interval $(0, 100)$
All the roots lie in the interval $[- 1, 99]$
None of these

The correct answer is: All the roots lie in the interval $open square brackets negative 1 comma blank 99 close square brackets$

$open parentheses x plus 1 close parentheses to the power of log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses end exponent equals 100 open parentheses x plus 1 close parentheses rightwards double arrow log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses to the power of log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses end exponent equals log subscript 10 end subscript invisible function application open parentheses 100 open parentheses x plus 1 close parentheses close parentheses$
$log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses equals 2 plus log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses$
Let $log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses equals y$
$rightwards double arrow y to the power of 2 end exponent minus y minus 2 equals 0 rightwards double arrow y equals 2 blank o r minus 1 rightwards double arrow log subscript 10 end subscript invisible function application open parentheses x plus 1 close parentheses equals 2 comma blank minus 1 rightwards double arrow x plus 1 equals 1001 divided by 10 rightwards double arrow x$
$equals 99 blank o r minus 9 divided by 10$

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