Question

please help me

two questions for answer

step by step​

Answer 1

$\rm \longrightarrow log_{ \sqrt{7} } \bigg( \dfrac{1}{343} \bigg) \\ \\ \\ \rm \longrightarrow log_{ \sqrt{7} } \bigg( \dfrac{1}{7 \times 7 \times 7} \bigg) \\ \\ \\ \rm \longrightarrow log_{ \sqrt{7} } \bigg( \dfrac{1}{ {7}^{3} } \bigg) \\ \\ \\ \rm \longrightarrow log_{ {7}^{ \frac{1}{2} } } ( {7})^{ - 3} \\ \\ \\ \rm \longrightarrow (- 3 \times 2) \: log_{ {7}^{ } } ( {7})\\ \\ \\ \rm \longrightarrow - 3 \times 2 \times 1 = - 6$

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$\rm \longrightarrow \dfrac{log_{ a }(125 ) }{log_{ a }(5 ) } \\ \\ \\ \rm \longrightarrow \dfrac{log_{ a } ({5}^{3} ) }{log_{ a }(5 ) } \\ \\ \\ \rm \longrightarrow 3 \times \dfrac{log_{ a } ({5}) }{log_{ a }(5 ) } \\ \\ \\ \rm \longrightarrow 3 \times 1 = 3$

Additional Information :-

$\boxed{\boxed{\begin{minipage}{5cm}\displaystyle\circ\sf\ ^{a} log \ a= 1\\\\\circ \ ^{a}log \ 1 = 0 \\\\\circ \ ^{a ^{n}} log \ b^{m}= \dfrac{m}{n} \times\:^{a}log \ b \\\\\circ \ ^{a^{m}} \ log \ b^{m} = \ ^{a}log \ b \\\\\circ \ ^{a}log \ b = \dfrac{1}{^{b}log \ a} \\\\\circ \ ^{a}log \ b = \dfrac{^{m}log \ b}{^{m} log \ a} \\\\\circ \ a^{^{a} logb} = b \\\\\circ \ ^{a}log \ b + ^{a}log \ c = \ ^{a}log(bc) \\\\\circ \ ^{a}log \ b -\: ^{a}log \ c = \ ^{a}log \left( \dfrac{b}{c} \right) \\\circ \ ^{a}log \ b \:\cdot\: ^{a}log \ c = \ ^{a}log \ c \\\\\circ \ ^{a}log \left( \dfrac{b}{c} \right) = \ ^{a}log \left(\dfrac{c}{b}\right)\end{minipage}}}</p><p></p><p></p><p>$

Answer 2

(viii) −3

Consider the law of logarithms

$Reminder = log_{ b }x = n ⇔ x = {b}^{n}$

$here b=7 \: and \: x= \frac{1}{343} \: n \: is \: the \: value \: of \: the \: logarithm.</p><p>$

$⇒1343=7n⇒7−3=7n⇒n=−3$

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(viii) 25

$\frac{125 \: log_{a} }{ log_{a} \times 5 }$

$\frac{125 \: log_{a} }{5 \: log_{a} }$

$\frac{125}{5}$

$25$

Hope This Helps