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Basic of Logarithms
medium
For $x \ne 0,{\left( {{{{x^l}} \over {{x^m}}}} \right)^{({l^2} + lm + {m^2})}}$${\left( {{{{x^m}} \over {{x^n}}}} \right)^{({m^2} + nm + {n^2})}}{\left( {{{{x^n}} \over {{x^l}}}} \right)^{({n^2} + nl + {l^2})}}=$
A
$1$
B
$x$
C
Does not exist
D
None of these
Solution
(a) ${\left( {{{{x^l}} \over {{x^m}}}} \right)^{{l^2} + lm + {m^2}}}{\left( {{{{x^m}} \over {{x^n}}}} \right)^{{m^2} + nm + {n^2}}}{\left( {{{{x^n}} \over {{x^l}}}} \right)^{{n^2} + nl + {l^2}}}$
$={({x^{l – m}})^{({l^2} + lm + {m^2})}}{({x^{m – n}})^{{m^2} + nm + {n^2}}}$${({x^{n – l}})^{{n^2} + nl + {l^2}}}$
$={x^{{l^3} – {m^3}}}.{x^{{m^3} – {n^3}}}.{x^{{n^3} – {l^3}}}$=${x^{{l^3} – {m^3} + {m^3} – {n^3} + {n^3} – {l^3}}} = {x^0}=1$
Standard 11
Mathematics
