open    $readhandle,    "<",    "Input.log";
open    $writehandle,    ">",    "Output.log";
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You should always verify that the files opened correctly:

open my $readhandle,  '<', 'Input.log'  or die "Cannot open 'Input.log
+' $!";
open my $writehandle, '>', 'Output.log' or die "Cannot open 'Output.lo
+g' $!";
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while(<$readhandle>)
{
$line    =    readline($readhandle);
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You are reading the first line into $_ and then reading the second line into $line so you only ever deal with half the lines in the file.   You probably want to work with every line of the file, like this:

while ( my $line = <$readhandle> )
{
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$line    =    ~/(.)\n/;
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You are assigning the bitwise negation of the match operator to $line.   It looks like you intended to use the binding operator (=~) instead although that expression would then be in void context and would not modify or output anything.