$P=\frac{4^a}{4^b}$, $Q=\frac{4^b}{4^c}$ , $R=\frac{4^c}{4^a}$

দেওয়া আছে, $R=\frac{4^c}{4^a}$

এখানে, log₄(R) = 0

বা, $lo{g}_4\frac{4^c}{4^a}=0$

বা, $lo{g}_4{4}^{c-a}=0$

বা, (c - a) log₄ (4) = 0

বা, (c - a) . l = 0

বা, c - a = 0

$\therefore$ a = c (দেখানো হলো)

দেওয়া আছে,

$P=\frac{4^a}{4^b}$, $Q=\frac{4^b}{4^c}$ , $R=\frac{4^c}{4^a}$

বামপক্ষ = $\sqrt{P}.\sqrt{Q}.\sqrt{R}$ = ${\left(\frac{4^a}{4^b}\right)}^{\frac{1}{ab}}{\left(\frac{4^b}{4^c}\right)}^{\frac{1}{bc}}{\left(\frac{4^c}{4^a}\right)}^{\frac{1}{ca}}$

= ${\left(4^{a-b}\right)}^{\frac{1}{ab}}{\left(4^{b-c}\right)}^{\frac{1}{bc}}{\left(4^{c-a}\right)}^{\frac{1}{ca}}$

= $4^{\frac{{}^{a-b}}{ab}}. 4^{\frac{b-c}{bc}}. 4^{\frac{c-a}{ca}}$

= $4^{\frac{{}^{a-b}}{ab}+\frac{b-c}{bc}+\frac{c-a}{ca}}$

$= 4^{\frac{ca - bc + ab - ca + bc - ab}{abc}}$

$=4^{\frac{0}{abc}}=4^0=1$

= ডানপক্ষ

$\sqrt[ab]{p}, \sqrt[bc]{Q}. \sqrt[ca]{R} = 1.$ (দেখানো হলো)

দেওয়া আছে

$P=\frac{4^a}{4^b}$, $Q=\frac{4^b}{4^c}$ , $R=\frac{4^c}{4^a}$

বামপক্ষ $(a^2+ab+b^2)logP+(b^2+bc+c^2)logQ+(c^2+ca+a^2) log\left(R\right)$

= $(a^2+ab+b^2)log \left(\frac{4^a}{4^b}\right)+(b^2+bc+c^2)log \left(\frac{4^b}{4^c}\right)+(c^2+ca+a^2) log \left(\frac{4^c}{4^a}\right)$

= $log {\left(\frac{4^a}{4^b}\right)}^{a^2+ab+b^2}+log {\left(\frac{4^b}{4^c}\right)}^{b^2+bc+c^2}+ log {\left(\frac{4^c}{4^a}\right)}^{c^2+ca+a^2}$

= $log \left\{4^{\left(a^3-b^3\right)}\times 4^{\left(b^3-c^3\right)}\times 4^{\left(c^3-a^3\right)}\right\}$

= $log \left(4^{a^3-b^3+b^3-c^3+c^3-a^3}\right)$

= $\log 4^0$

= $\log 1$

= 0

= ডানপক্ষ

$(a^2+ab+b^2)logP+(b^2+bc+c^2)logQ+(c^2+ca+a^2) log\left(R\right)$ = 0 (প্রমাণিত)