How to prove that x*y*z=1 if a^x=b, b^y=c, c^z=a and a>0, b>0, c>0 and none of them can equal 1?

1 Answer
Tony B
Oct 8, 2017

As soon as you see the format type #color(brown)(b^y=c)# consider if you can use logs. Perhaps you can and perhaps not. In this case, yes!

Explanation:

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#

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