Let,$x^{12}=y^{16}=z^{24}=k$
So,$\log x=\dfrac{\log k}{12}$
$\log y=\dfrac{\log k}{16}$
$\log z=\dfrac{\log k}{24}$
Now, it is given,$3\log_{y}x, 4 \log_{z}y, n\log_{x}z$ are in A.P.
So,$2\cdot\left(4\log_zy\right)=3\log_yx+n\log_xz$ -------(1)
or,$\dfrac{8\log y}{\log z}=\dfrac{3\log x}{\log y}+\dfrac{n\log z}{\log x}$
Now,$\dfrac{\log y}{\log z}=\dfrac{\dfrac{\log k}{16}}{\dfrac{\log k}{24}}=\dfrac{24}{16}=\dfrac{3}{2}$
Also,$\dfrac{\log x}{\log y}=\dfrac{\dfrac{\log k}{12}}{\dfrac{\log k}{16}}=\dfrac{16}{12}=\dfrac{4}{3}$
also,$\dfrac{\log z}{\log x}=\dfrac{\dfrac{\log k}{24}}{\dfrac{\log k}{12}}=\dfrac{12}{24}=\dfrac{1}{2}$
So, we can rewrite equation (1) as,
$8\left(\dfrac{3}{2}\right)=3\left(\dfrac{4}{3}\right)+n\left(\dfrac{1}{2}\right)$
or,$12=4+\dfrac{n}{2}$
or,$n=16$
