Question

Find the values of

$(a) \: log_{25}(200)$
$(b) \: log_{7}( \sqrt{2} )$
$(c) \: log( \sqrt[3]{48} \times {108}^{ \frac{1}{4} } \div \sqrt[12]{6} )$


❌❌NO SPAMMING❌❌​

Answer 1

Logarithmic Table :

$\begin{array}{|c|c|}\cline{1-2}\mathrm{log_{e}2} & 0.693\\ \cline{1-2}\mathrm{log_{e}5} & 1.609\\ \cline{1-2}\mathrm{log_{e}7}&1.946\\ \cline{1-2}\mathrm{log_{10}2}&0.301\\ \cline{1-2}\mathrm{log_{10}3}&0.477\\ \cline{1-2}\end{array}$

(a) Now, $\mathrm{log_{25}200}$

$\mathrm{=\frac{log_{e}200}{log_{e}25}}$

$\mathrm{=\frac{log_{e}(5^{2}\times 2^{3})}{log_{e}(5^{2})}}$

$\mathrm{=\frac{log_{e}5^{2}+log_{e}2^{3}}{log_{e}5^{2}}}$

$\mathrm{=\frac{log_{e}5^{2}}{log_{e}5^{2}}+\frac{log_{e}2^{3}}{log_{e}5^{2}}}$

$\mathrm{=1+\frac{3log_{e}2}{2log_{e}5}}$

$\approx \mathrm{1+\frac{3\times 0.693}{2\times 1.609}}$

$\approx \mathrm{1.646}$

$\to \boxed{\mathrm{\frac{log_{e}200}{log_{e}25}\approx 1.646}}$

(b) Now, $\mathrm{log_{7}\sqrt{2}}$

$\mathrm{=log_{7}2^{\frac{1}{2}}}$

$\mathrm{=\frac{1}{2}log_{7}2}$

$\mathrm{=\frac{1}{2}\frac{log_{e}7}{log_{e}2}}$

$\approx \mathrm{\frac{1}{2}\frac{0.693}{1.946}}$

$\approx \mathrm{0.178}$

$\to \boxed{\mathrm{log_{7}\sqrt{2}\approx 0.178}}$

(c) Now, $\mathrm{log(\sqrt[3]{48}\times 108^{\frac{1}{4}}\div \sqrt[12]{6})}$

$\mathrm{=log\bigg(\frac{48^{\frac{1}{3}}\times 108^{\frac{1}{4}}}{6^{\frac{1}{12}}}\bigg)}$

$\mathrm{=log48^{\frac{1}{3}}+log108^{\frac{1}{4}}-log6^{\frac{1}{12}}}$

$\mathrm{=\frac{1}{3}log(2^{4}\times 3)+\frac{1}{4}log(2^{2}\times 3^{3})-\frac{1}{12}log(2\times 3)}$

$\mathrm{=\frac{4}{3}log2+\frac{1}{3}log3+\frac{1}{2}log2+\frac{3}{4}log3-\frac{1}{12}log2-\frac{1}{12}log3}$

$\mathrm{=\frac{7}{4}log_{10}2+log_{10}3}$

$\approx \mathrm{\frac{7}{4}(0.301)+0.477}$

$\approx \mathrm{1.00375}$

$\to \boxed{\mathrm{log(\sqrt[3]{48}\times 108^{\frac{1}{4}}\div \sqrt[12]{6})\approx 1.00375}}$

Answer 2

Answer:

$\huge\underline\mathfrak\green{Solution}$