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Question

If $log subscript 2 end subscript invisible function application open parentheses 5 cross times 2 to the power of x end exponent plus 1 close parentheses comma log subscript 4 end subscript invisible function application left parenthesis 2 to the power of 1 minus x end exponent plus 1 right parenthesis$ and 1 are in A.P., then $x$ equals

$\log_{2} 5$
$1 - \log_{5} 2$
$\log_{5} 2$
None of these

The correct answer is: None of these

The given numbers are in A.P. Therefore,
$2 log subscript 4 end subscript invisible function application left parenthesis 2 to the power of 1 minus x end exponent plus 1 right parenthesis equals log subscript 2 end subscript invisible function application left parenthesis 5 cross times 2 to the power of x end exponent plus 1 right parenthesis plus 1$
$rightwards double arrow 2 log subscript 2 to the power of 2 end exponent end subscript invisible function application open parentheses fraction numerator 2 over denominator 2 to the power of x end exponent end fraction plus 1 close parentheses equals log subscript 2 end subscript invisible function application open parentheses 5 cross times 2 to the power of x end exponent plus 1 close parentheses plus log subscript 2 end subscript invisible function application 2$
$rightwards double arrow fraction numerator 2 over denominator 2 end fraction log subscript 2 end subscript invisible function application open parentheses fraction numerator 2 over denominator 2 to the power of x end exponent end fraction plus 1 close parentheses equals log subscript 2 end subscript invisible function application open parentheses 5 cross times 2 to the power of x end exponent plus 1 close parentheses 2$
$rightwards double arrow log subscript 2 end subscript invisible function application open parentheses fraction numerator 2 over denominator 2 to the power of x end exponent end fraction plus 1 close parentheses equals log subscript 2 end subscript invisible function application open parentheses 10 cross times 2 to the power of x end exponent plus 2 close parentheses$
$rightwards double arrow blank fraction numerator 2 over denominator 2 to the power of x end exponent end fraction plus 1 equals 10 cross times 2 to the power of x end exponent plus 2$
$rightwards double arrow fraction numerator 2 over denominator y end fraction plus 1 equals 10 y plus 2$ , where $2 to the power of x end exponent equals y$
$rightwards double arrow 10 y to the power of 2 end exponent plus y minus 2 equals 0$
$rightwards double arrow open parentheses 5 y minus 2 close parentheses open parentheses 2 y plus 1 close parentheses equals 0$
$rightwards double arrow blank y equals 2 divided by 5$ or $y equals negative 1 divided by 2$
$rightwards double arrow blank 2 to the power of x end exponent equals 2 divided by 5$ or $2 to the power of x end exponent equals negative 1 divided by 2$
$rightwards double arrow x equals log subscript 2 end subscript invisible function application open parentheses 2 divided by 5 close parentheses blank left square bracket because 2 to the power of x end exponent$ cannot be negative]
$rightwards double arrow x equals log subscript 2 end subscript invisible function application 2 minus log subscript 2 end subscript invisible function application 5$
$rightwards double arrow x equals 1 minus log subscript 2 end subscript invisible function application 5$

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