If log_a b + log_b c + log_c a vanishes,where | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $x = \log_a b$,$y = \log_b c$,and $z = \log_c a$.Given that $x + y + z = 0$.We know the algebraic identity: if $x + y + z = 0$,then $x^3 + y^3

Vedclass · Accepted Answer

an odd prime

Vedclass · Answer

(A) Let $x = \log_a b$,$y = \log_b c$,and $z = \log_c a$.Given that $x + y + z = 0$.We know the algebraic identity: if $x + y + z = 0$,then $x^3 + y^3 + z^3 = 3xyz$.Now,consider the product $xyz = (\log_a b) \cdot (\log_b c) \cdot (\log_c a)$.Using the change of base formula,$\log_a b = \frac{\ln b}{\ln a}$,$\log_b c = \frac{\ln c}{\ln b}$,and $\log_c a = \frac{\ln a}{\ln c}$.Thus,$xyz = \left( \frac{\ln b}{\ln a} \right) \cdot \left( \frac{\ln c}{\ln b} \right) \cdot \left( \frac{\ln a}{\ln c} \right) = 1$.Therefore,$x^3 + y^3 + z^3 = 3(1) = 3$.Since $3$ is an odd prime number,the correct option is $A$.

If $\log_a b + \log_b c + \log_c a$ vanishes,where $a, b,$ and $c$ are positive real numbers different from unity,then the value of $(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3$ is

Similar Questions

The value of $\frac{\sin^2 \theta}{\sin \theta - \cos \theta} - \frac{\sin \theta + \cos \theta}{\tan^2 \theta - 1}$ for all permissible values of $\theta$ is:

$\cos^2 48^\circ - \sin^2 12^\circ = $

If $\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1$,then the value of $\left[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} \right]$ is (where $[.]$ denotes the greatest integer function).

If $A + B + C = 180^o$,then the value of $\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2}$ is equal to

The circular measure of an angle of an isosceles triangle is $\frac{5 \pi}{9}$. The circular measure of one of the other angles must be

Vedclass Test Series

Exam Paper Generator

Online Exam Module