৫ $x=2$, $y=3$, $z=5$, $w=7$
- (ক) $\sqrt{y^3}$ এর $3$ ভিত্তিক লগ নির্ণয় কর।
- (খ) $w\log\dfrac{xz}{y^2}-x\log\dfrac{z^2}{x^2y}+y\log\dfrac{y^4}{x^4z}$ এর মান নির্ণয় কর।
- (গ) দেওখাও যে, $\dfrac{\log\sqrt{y^3}+y\log x-\frac{y}{x}\log\left(xz\right)}{\log\left(xy\right)-\log z}=\log_{y}\sqrt{y^3}$
(ক) নং এর সমাধান
দেওয়া আছে, $y=3$
$\therefore$ $\sqrt{y^3}$ এর $3$ ভিত্তিক লগ
$=\log_3\sqrt{y^3}$
$=\log_3\sqrt{3^3}$
$=\log_33^{3\cdot \frac12}$
$=\frac32\log_33$
$=\frac32\cdot 1$
$=\frac32$ [Answer]
(খ) নং এর সমাধান
দেওয়া আছে, $x=2$, $y=3$, $z=5$, $w=7$
প্রদত্ত রাশি
$w\log\dfrac{xz}{y^2}-x\log\dfrac{z^2}{x^2y}+y\log\dfrac{y^4}{x^4z}$
$=7\log\dfrac{xz}{y^2}-2\log\dfrac{z^2}{x^2y}+3\log\dfrac{y^4}{x^4z}$
$=\log\left(\dfrac{xz}{y^2}\right)^7-\log\left(\dfrac{z^2}{x^2y}\right)^2+\log\left(\dfrac{y^4}{x^4z}\right)^3$
$=\log\dfrac{x^7z^7}{y^{14}}-\log\dfrac{z^4}{x^4y^2}+\log\dfrac{y^{12}}{x^{12}z^3}$
$=\log\left(\dfrac{x^7z^7}{y^{14}}\div\dfrac{z^4}{x^4y^2}\times\dfrac{y^{12}}{x^{12}z^3}\right)$
$=\log\left(\dfrac{x^7z^7}{y^{14}}\times\dfrac{x^4y^2}{z^4}\times\dfrac{y^{12}}{x^{12}z^3}\right)$
$=\log\left(\dfrac{x^{11}y^{14}z^{7}}{x^{12}y^{14}z^{7}}\right)$
$=\log\left(\dfrac{1}{x}\right)$
$=\log\left(\dfrac12\right)$
$=\log\left(2^{-1}\right)$
$=-1\log2$
$=-\log2$ [Answer]
(গ) নং এর সমাধান
দেওয়া আছে, $x=2$, $y=3$, $z=5$, $w=7$
Left Hand Side,
$\dfrac{\log\sqrt{y^3}+y\log x-\frac{y}{x}\log\left(xz\right)}{\log\left(xy\right)-\log z}$
$=\dfrac{\log\sqrt{3^3}+3\log2-\frac{3}{2}\log\left(2\times5\right)}{\log\left(2\times3\right)-\log5}$
$=\dfrac{\log\sqrt{3^3}+3\log2-\left(\frac32\log2+\frac32\log5\right)}{\left(\log2+\log3\right)-\log5}$
$=\dfrac{\log\sqrt{3^3}+3\log2-\frac32\log2-\frac32\log5}{\log2+\log3-\log5}$
$=\dfrac{\log3^{3\times\frac12}+3\log2-\frac32\log2-\frac32\log5}{\log2+\log3-\log5}$
$=\dfrac{\log3^{\frac32}+\left(3-\frac32\right)\log2-\frac32\log5}{\log2+\log3-\log5}$
$=\dfrac{\frac32\log3+\left(\frac{6-3}{2}\right)\log2-\frac32\log5}{\log2+\log3-\log5}$
$=\dfrac{\frac32\log3+\frac32\log2-\frac32\log5}{\log2+\log3-\log5}$
$=\dfrac{\frac32\left(\log3+\log2-\log5\right)}{\log2+\log3-\log5}$
$=\frac32$
Right Hand Side,
$\log_{y}\sqrt{y^3}$
$=\log_{3}\sqrt{3^3}$
$=\log_33^{3\times\frac12}$
$=\log_33^{\frac32}$
$=\frac32\log_33$
$=\frac32\times1$
$=\frac32$
$\therefore$ Left Hand Side $=$ Right Hand Side
সুতরাং, $x=2$, $y=3$, $z=5$, $w=7$ হলে $\dfrac{\log\sqrt{y^3}+y\log x-\frac{y}{x}\log\left(xz\right)}{\log\left(xy\right)-\log z}=$$\log_{y}\sqrt{y^3}$ [Showed]