Logarithms!

Algebra Level 4

log2x+log4y+log4z=2log3y+log9z+log9x=2log4z+log16x+log16y=2\large \begin{aligned} \log_2 x + \log_4 y + \log_4 z &=& 2 \\ \log_3 y + \log_9 z + \log_9 x &=& 2 \\ \log_4 z + \log_{16} x + \log_{16} y &=& 2 \end{aligned}

Solve the system equations where xx , yy and zz are positive real numbers.

If the value x,y,zx,y,z can be written in the form axbx,ayby,azbz\dfrac{a_x}{b_x} , \dfrac{a_y}{b_y} , \dfrac{a_z}{b_z} for co-prime ai,aja_i , a_j, find the value of ax+bx+ay+by+az+bza_x+b_x+a_y+b_y+a_z+b_z.

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