We can use any logs we so wish for this. I choose #log_e->ln#
#a^x=b color(white)("d")->color(white)("d")xln(a)=ln(b)" "....Equation(1)#
#b^y=c color(white)("d")-> color(white)("d")yln(b)=ln(c)" ".......Equation(2)#
#c^z=a color(white)("d")->color(white)("d")zln(c)=ln(a)" "...Equation(3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From #Equation(3) color(white)("d")ln(c)=ln(a)/z" "..Eqn(3_a)#
Thus #Equation(2)# becomes #color(white)("d")yln(b)=ln(a)/z#
Thus #color(white)("ddddddddddd")zyln(b)=ln(a)" "..Eqn(2_a)#
But from #Equation(1)color(white)("d") ln(a)=ln(b)/x#
Substitute into #Eqn(2_a)#
#color(white)("dddddddddddddd")zyln(b)=ln(b)/x#
#color(white)("dddddddddddddd")=> zyx=ln(b)/ln(b)=1#