0

If $ \log_p q + \log_q r + \log_r p = 0 $

Then what is the value of,

$$(\log_p q)^3 + (\log_q r)^3 + (\log_r p)^3$$

given that $p,q,r \neq 1$

A. It is odd prime

B. It is even prime

C. Odd composite

D. Irrational

I have tried using the identity that, if $ a + b + c = 0$, $a^3+b^3+c^3= 3abc$ , but it gives me the answer $0$.

Community
  • 1
Fghj
  • 1,471

2 Answers2

2

HINT


$x^3+y^3+z^3=3xyz$ when $x+y+z=0$.

AND


$\log_p q=\frac{\log q}{\log p}$, $\log_q r=\frac{\log r}{\log q}$, $\log_r p=\frac{\log p}{\log r}$.

Tianlalu
  • 5,177
1

Based on the identity

$$a+b+c=0 \Rightarrow a^3+b^3+c^3 =3abc,$$ the answer is 3. Just note that

$\log_p(q)=\frac{ln(q)}{ln(p)},$ etc, so your hypothesis is just

$$\frac{ln(q)}{ln(p)}+\frac{ln(r)}{ln(q)}+\frac{ln(p)}{ln(r)}=0.$$ Put now

$$a= \frac{ln(q)}{ln(p)} , \; b=\frac{ln(r)}{ln(q)},\; c=\frac{ln(p)}{ln(r)}$$ and note that $abc=1.$

Hope this helps

John D
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