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
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