Question:
If $2^{x}=3^{y}=6$, show that $1 / x+1 / y+1 / z=0$
Solution:
$2^{x}=3^{y}=6^{-z}$
$2^{x}=k$
$2=k^{1 / x}$
$3^{y}=k$
$3=k^{1 / y}$
$6^{-z}=k$
$k=1 / 6^{z}$
$6=k^{-1 / z}$
$2 \times 3=6$
$k^{1 / x} \times k^{1 / y}=k^{-1 / z}$
$1 / x+1 / y=-1 / z$ [by equating exponents]
$1 / x+1 / y+1 / z=0$
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