Question

(log root 27 + log 8 + log root 1000)÷ log 120

Answer 1

ANSWER:

$\small{ \sf{ \frac{log \: \sqrt{27} + log \: 8 + log \sqrt{1000} }{log \: 120} }}$

$\small {\sf{ = \frac{log3 \sqrt{3} + log8 + log10. {10}^{ \frac{1}{2} } }{log120} }}$

$\small {\sf{ = \frac{log3 \sqrt{3} + log8 + log {10}^{ \frac{3}{2} } }{log120}}}$

$= \small{ \sf{ \frac{log3 \sqrt{3 }+ log8 + \frac{3}{2}log10 } {log120} }}$

Using log_a (a) = 1

$= \small{ \sf{ \frac{log3 \sqrt{3} + log8 + \frac{3}{2} }{log120} }}$

write all the numerators above the common denominator

$\small{ \sf{ = \frac{2log3 \sqrt{3} +2 log8 + 3 }{2log120} }}$

We know that

$\small{ \sf{2log \: a = log {a}^{2} }}$

$\small{ \sf{ = \frac{log {(3 \sqrt{3}) }^{2} + log {8}^{2} + {3} }{2log120} }}$

$\small{ \sf{ = \frac{log27 + log64 + {3} }{2log120} }}$

$\small{ \sf{ = \frac{log27 + log64 + 3log10}{2log120} }}$

$\small { \sf{= \frac{log27 + log64 + log {10}^{3} }{2log120} }}$

Using

log a + log b + log c = log(abc)

$\small{ \sf{ \frac{log(27 \times 64 \times {10}^{3}) }{2log120} = \frac{1}{2} \times \frac{log(27 \times 64 \times {10}^{3} )}{log120} }}$

Using

$\frac{ \log \: a}{ \log \: b} = \log_{b}(a)$

$\small{ \sf{ = \frac{1}{2} \times {log_{120} (1728 \times {10}^{3}) } }}$

$\small{ \sf{ \to \: \frac{ log_{120} {120}^{3} }{2} = \frac{3}{2} }}$

Hence,

$\to \small{ \bf{ \frac{log \sqrt{27} + log8 + log \sqrt{1000} }{log120} = \frac{3}{2} }}$

Answer 2

$\dfrac{log3 \sqrt{3} + log8 + log10. {10}^{ \frac{1}{2} } }{log120} } \\ \\\dfrac{log3 \sqrt{3} + log8 + log {10}^{ \frac{3}{2} } }{log120} \\ \\ \dfrac{log3 \sqrt{3 }+ log8 + \frac{3}{2}log10 } {log120} \\\\\dfrac{log3 \sqrt{3} + log8 + \frac{3}{2} }{log120} \\ \\ \dfrac{2(log3 \sqrt{3} + log8 + \frac{3}{2}) }{ \frac{2}{log120} }\\ \\</p><p>\dfrac{2log3 \sqrt{3} +2 log8 + 3 }{2log120} } \\ \\\dfrac{log {(3 \sqrt{3}) }^{2} + log {8}^{2} + {3} }{2log120} \\ \\ \dfrac{log27 + log64 + {3} }{2log120} \\ \dfrac{log27 + log64 + 3log10}{2log120} \\ \\ \dfrac{log \sqrt{27} + log8 + log \sqrt{1000} }{log120} \\ \\ = \dfrac{3}{2}$