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a = log{ 12 base 24 }
b = log{ 24 base 36 }
c = log{ 36 base 48 }
we know ,
log{ a base b } = log{ a base e }/log{ b base e }
use this concept here,
now,
abc = log{ 12 base 24} × log{ 24 base 36 } × log{ 36 base 48 }
= log12/log24 × log24/log36 × log36/log48
= log12/log48
now,
LHS 1 + abc = 1 + log12/log48
=log48/log48 + log12/log48
=( log48 + log12)/log48
=log(48 × 12)/log48
=log( 24 × 24 )/log48
= 2log24/log48
= 2{ log24/log36 }{ log36/log48}
= 2log{ 24 base 36 }.log{ 36 base 48 }
=2bc
b = log{ 24 base 36 }
c = log{ 36 base 48 }
we know ,
log{ a base b } = log{ a base e }/log{ b base e }
use this concept here,
now,
abc = log{ 12 base 24} × log{ 24 base 36 } × log{ 36 base 48 }
= log12/log24 × log24/log36 × log36/log48
= log12/log48
now,
LHS 1 + abc = 1 + log12/log48
=log48/log48 + log12/log48
=( log48 + log12)/log48
=log(48 × 12)/log48
=log( 24 × 24 )/log48
= 2log24/log48
= 2{ log24/log36 }{ log36/log48}
= 2log{ 24 base 36 }.log{ 36 base 48 }
=2bc
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