Question:
If $2^{x}=3^{y}=12^{z}$, show that $1 / z=1 / y+2 / x$
Solution:
$2^{x}=3^{y}=(2 \times 3 \times 2)^{z}$
$2^{x}=3^{y}=\left(2^{2} \times 3\right)^{z}$
$2^{x}=3^{y}=\left(2^{2 z} \times 3^{z}\right)$
$2^{x}=3^{y}=12^{z}=k$
$2=k^{1 / x}$
$3=k^{1 / y}$
$12=k^{1 / z}$
$12=2 \times 3 \times 2$
$12=k^{1 / z}=k^{1 / y} \times k^{1 / x} \times k^{1 / x}$
$k^{1 / z}=k^{2 / x}+{ }^{1 / y}$
$1 / z=1 / y+2 / x$
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