$log_2 x$ + $log_4 y$ + $log_4 z$ =2

$\frac{log x}{log 2}$ + $\frac{log y}{log 2^{2}}$ + $\frac{log z}{log 2^{2}}$ =2

$\frac{log x}{log 2}$ + $\frac{log y}{2log 2}$ + $\frac{log z}{2log 2}$ =2

Multiply both sides by $2log 2$ .

$2log x$ + $log y$ + $log z$ = $4log 2$

$log x^{2}$ + $log y$ + $log z$ = $4log 2$

$log x^{2}yz$ = $log 2^{4}$

$x^{2}yz=2^{4}$ (Equation No. 1)

$log_3 y$ + $log_9 x$ + $log_9 z$ =2

$\frac{log y}{log 3}$ + $\frac{log x}{log 3^{2}}$ + $\frac{log z}{log 3^{2}}$ =2

$\frac{log y}{log 3}$ + $\frac{log x}{2log 3}$ + $\frac{log z}{2log 3}$ =2

Multiply both sides by $2log 3$ .

$2log y$ + $log x$ + $log z$ = $4log 3$

$log y^{2}$ + $log x$ + $log z$ = $4log 3$

$log xy^{2}z$ = $log 3^{4}$

$xy^{2}z=3^{4}$ (Equation No. 2)

$log_4 z$ + $log_16 x$ + $log_16 y$ =2

$\frac{log z}{log 2^{2}}$ + $\frac{log x}{log 2^{4}}$ + $\frac{log y}{log 2^{4}}$ =2

$\frac{log z}{2log 2}$ + $\frac{log x}{4log 2}$ + $\frac{log y}{4log 2}$ =2

Multiply both sides by $4log 2$ .

$2log z$ + $log x$ + $log y$ = $8log 2$

$log z^{2}$ + $log x$ + $log y$ = $8log 2$

$log xyz^{2}$ = $log 2^{8}$

$xyz^{2}=2^{8}$ (Equation No. 3)

Multiply Equations 1, 2 and 3.

$x^{4}y^{4}z^{4}=2^{4}3^{4}2^{8}$

$x^{4}y^{4}z^{4}=2^{12}3^{4}$

$xyz=2^{3}3$

or

$xyz=-2^{3}3$ -Extraneous Solution-

$xyz=24$ (ANSWER)