Lugging logs

Algebra Level 4

Let $a, b, c > 1$ and $x, y, z$ be reals such that $a^x = b, b^y = c$ and $c^z = a$ .

As the values of $a$ , $b$ , and $c$ change in the domain above, what is the minimum value of $\\ x + y + z$ ?

1 solution

Chew-Seong Cheong
Jul 29, 2020

Given that $\begin{cases} a^x = b \\ b^y = c \\ c^z = a \end{cases}$ $\implies b^y = a^{xy} = c^{xyz} = c \implies xyz = 1$ .

From AM-GM inequality $x + y + z \ge 3\sqrt[3]{xyz} = \boxed 3$ . Equality occurs when $a=b=c$ .

Wow that's pretty elegant compared to my solution :)

On a side note, equality holds when a = b = c only.

Steven Jim - 10 months, 2 weeks ago

Yes, thanks. I will change it.

Chew-Seong Cheong - 10 months, 2 weeks ago

You should say that a, b, c>1 because otherwise for a, b, c =1 you have a free choice for x, y and z

Matteo Bianchi - 10 months, 2 weeks ago

Updated. Thanks!

Steven Jim - 10 months, 1 week ago