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)
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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