Question

यदि log10 2=0.3010और log10 3=0.4771 हो तो मान ज्ञात कीजिए log10√8​

Answer 1

Given:

$\rm log_{10}2 = 0.3010 \\ \rm log_{10}3 = 0.4771$

To Find:

$\rm log_{10} \sqrt{8}$

Answer:

$\rm \implies log_{10} \sqrt{8} \\ \\ \rm \implies log_{10} {8}^{ \frac{1}{2} } \\ \\ \rm \implies log_{10} {2}^{ \frac{3}{2} } \\ \\ \rm \implies \dfrac{3}{2} log_{10}2 \\ \\ \rm \implies \dfrac{3}{2} \times 0.3010 \\ \\ \rm \implies 1.5 \times 0.3010 \\ \\ \rm \implies 0.4515$

$\therefore$ $\boxed{\mathfrak{log_{10} \sqrt{8} = 0.4515}}$

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

ᴠᴀʟᴜᴇ ᴏꜰ ʟᴏɢ₁₀√8​

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\bf{\log _{10}\left(\sqrt{8}\right)=\frac{3}{2}\log _{10}\left(2\right)\quad \left(\mathrm{Decimal:\quad }\:0.45154\dots \right)}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ :

_____________________________

→ ꜰɪʀꜱᴛ ꜱᴏʟᴠᴇ ꜰᴏʀ $\bf{\sqrt{8}}$

$\mathrm{Prime\:factorization\:of\:}8:\quad 2^3$

$\sqrt{8}=\sqrt{2^3}$

$\mathrm{Apply\:exponent\:rule}:\quad \:a^{b+c}=a^b\cdot \:a^c$

$2^3=2^2\cdot \:2$

$=\sqrt{2^2\cdot \:2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \:a\ge 0,\:b\ge 0$

$\sqrt{2^2\cdot \:2}=\sqrt{2^2}\sqrt{2}$

$=\sqrt{2^2}\sqrt{2}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt{a^2}=a,\:\quad \:a\ge 0$

$\sqrt{2^2}=2$

$=2\sqrt{2}$

$\bf{\log _{10}\left(\sqrt{8}\right) =\log _{10}\left(2\sqrt{2}\right)}$

_____________________________

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(xy\right)=\log _a\left(x\right)+\log _a\left(y\right)$

$\log _{10}\left(2\sqrt{2}\right)=\log _{10}\left(2\right)+\log _{10}\left(\sqrt{2}\right)$

$=\log _{10}\left(2\right)+\log _{10}\left(\sqrt{2}\right)$

_____________________________

$\log _{10}\left(\sqrt{2}\right)=\frac{1}{2}\log _{10}\left(2\right)$

$\mathrm{Rewrite\:\log _{10}\left(\sqrt{2}\right)\:as}$

$=\log _{10}\left(2^{\frac{1}{2}}\right)$

$\mathrm{Apply\:log\:rule\:}\log _a\left(x^b\right)=b\cdot \log _a\left(x\right),\:\quad \mathrm{\:assuming\:}x\:\ge \:0$

$=\frac{1}{2}\log _{10}\left(2\right)$

$\bf{=\log _{10}\left(2\right)+\frac{1}{2}\log _{10}\left(2\right)}$

_____________________________

$\mathrm{Factor\:out\:common\:term\:}\log _{10}\left(2\right)$

$=\log _{10}\left(2\right)\left(1+\frac{1}{2}\right)$

_____________________________

$1+\frac{1}{2}=\frac{3}{2}$

$=\frac{3}{2}\log _{10}\left(2\right)$

_____________________________

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}$

$=\frac{3\log _{10}\left(2\right)}{2}$

_____________________________

$\boxed{\bf{= 0.45154\dots }}$