यदि घनाभ की भुजाएँ V₁ = log_abc a³, V₂ = log_abc b³ और V₃ = logabc c³ हैं, तो ${\mathrm{V}}_1^2+{\mathrm{V}}_2^2+{\mathrm{V}}_3^2+2({\mathrm{V}}_1{\mathrm{V}}_2+{\mathrm{V}}_2{\mathrm{V}}_3+{\mathrm{V}}_1{\mathrm{V}}_3)$ का मान कितना है?

दिया गया है:

V1 = logabc a3, V2 = logabc b3 और V3 = logabc c3  

${\mathrm{V}}_1^2+{\mathrm{V}}_2^2+{\mathrm{V}}_3^2+2({\mathrm{V}}_1{\mathrm{V}}_2+{\mathrm{V}}_2{\mathrm{V}}_3+{\mathrm{V}}_1{\mathrm{V}}_3)$ = ?

प्रयुक्त अवधारणा:

Log_ab = ${\displaystyle \frac{lo{g}_{c}b}{lo{g}_{c}a}}$

log₁₀abc = log₁₀a + log₁₀b + log₁₀c

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

गणना:

V1 = logabc a3 = ${\displaystyle \frac{lo{g}_{10}{a}^3}{lo{g}_{10}abc}}$ = ${\displaystyle \frac{3lo{g}_{10}a}{lo{g}_{10}abc}}$ 

V2 = logabc b3 = ${\displaystyle \frac{lo{g}_{10}{b}^3}{lo{g}_{10}abc}}$ = ${\displaystyle \frac{3lo{g}_{10}b}{lo{g}_{10}abc}}$

V3 = logabc c3 = ${\displaystyle \frac{lo{g}_{10}{c}^3}{lo{g}_{10}abc}}$ = ${\displaystyle \frac{3lo{g}_{10}c}{lo{g}_{10}abc}}$

V₁ + V₂ + V₃ = ${\displaystyle \frac{3lo{g}_{10}a}{lo{g}_{10}abc}}$ + ${\displaystyle \frac{3lo{g}_{10}b}{lo{g}_{10}abc}}$ + ${\displaystyle \frac{3lo{g}_{10}c}{lo{g}_{10}abc}}$

⇒ 3 × ${\displaystyle \frac{lo{g}_{10}a+lo{g}_{10}b+lo{g}_{10}c}{lo{g}_{10}a+lo{g}_{10}b+lo{g}_{10}c}}$ = 3

अब, ${\mathrm{V}}_1^2+{\mathrm{V}}_2^2+{\mathrm{V}}_3^2+2({\mathrm{V}}_1{\mathrm{V}}_2+{\mathrm{V}}_2{\mathrm{V}}_3+{\mathrm{V}}_1{\mathrm{V}}_3)$ 

⇒ (V1 + V2 + V3)² = 3² = 9

∴ ${\mathrm{V}}_1^2+{\mathrm{V}}_2^2+{\mathrm{V}}_3^2+2({\mathrm{V}}_1{\mathrm{V}}_2+{\mathrm{V}}_2{\mathrm{V}}_3+{\mathrm{V}}_1{\mathrm{V}}_3)$ = 9