Question

chapter logarithm solution need​

Answer 1

Given Question :- Evaluate

$\rm \: log_{3 \sqrt{2} }(324)$

$\large\underline{\sf{Solution-}}$

Given logarithmic expression is

$\rm \: log_{3 \sqrt{2} }(324)$

Let first find the prime factors of 324.

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:324\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:162 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:81\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

So,

$\rm \: 324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$

can be further rewritten as

$\rm \: 324 = \sqrt{2} \times \sqrt{2} \times \sqrt{2} \times \sqrt{2} \times 3 \times 3 \times 3 \times 3$

$\rm\implies \:324 = {(3 \sqrt{2}) }^{4}$

Now,

$\rm \: log_{3 \sqrt{2} }(324)$

can be rewritten as

$\rm \: = \: log_{3 \sqrt{2} }[{(3 \sqrt{2})}^{4}]$

We know,

$\color{green}\boxed{ \rm{ \: \: \: log_{x}( {x}^{y} ) = y \: \: \: }}$

So, using this result, we get

$\rm \: = \: 4$

Hence,

$\color{green}\rm\implies \:\boxed{ \rm{ \:\rm \: log_{3 \sqrt{2} }(324) \: = \: 4 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ log_{x}(x) = 1}\\ \\ \bigstar \: \bf{ log_{x}( {x}^{y} ) = y}\\ \\ \bigstar \: \bf{ log_{ {x}^{z} }( {x}^{w} ) = \dfrac{w}{z} }\\ \\ \bigstar \: \bf{ log_{a}(b) = \dfrac{logb}{loga} }\\ \\ \bigstar \: \bf{ {e}^{logx} = x}\\ \\ \bigstar \: \bf{ {e}^{ylogx} = {x}^{y}}\\ \\ \bigstar \: \bf{log1 = 0}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer:

your solution is in above pic

pls mark it as brainliest