Question

Please help solve this logarithm problem.
What is the value of the number given below? (please show step-by-step process)

Answer 1

Given Question

Evaluate

$\rm :\longmapsto\: log_{ \sqrt{3} }(729)$

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

Given Logarithmic expression is

$\rm :\longmapsto\: log_{ \sqrt{3} }(729)$

Let first factorize 729

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:729 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:81\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:}}\\\underline{\sf{}}&{\sf{\:\:3 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

So,

$\purple{\rm :\longmapsto\:729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3}$

$\purple{\rm :\longmapsto\:729 = {3}^{6} }$

$\purple{\rm :\longmapsto\:729 = {( [{ \sqrt{3} ]}^{2}) }^{6} }$

$\purple{\rm :\longmapsto\:729 = {( \sqrt{3} )}^{12}}$

So,

$\rm :\longmapsto\: log_{ \sqrt{3} }(729)$

can be rewritten as

$\rm \:  =  \: log_{ \sqrt{3} }( {( \sqrt{3}) }^{12} )$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ log_{a}( {a}^{x} ) \: = \: x \: }}}$

So, using this identity, we get

$\rm \:  =  \: 12$

Hence,

$\green{\rm\implies \:\boxed{\tt{ \: \: log_{ \sqrt{3} }(729) \: = \: 12 \: \: }}}$

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More to Know

$\purple{\rm :\longmapsto\:\boxed{\tt{ log_{x}(x) = 1 \: }}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ log_{ {x}^{y} }( {x}^{z} ) = \frac{z}{y} \: }}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ log_{ {a}^{y} }( {b}^{z} ) = \frac{z}{y} log_{a}(b) \: }}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ {a}^{ log_{a}(x) } = x}}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ {a}^{ ylog_{a}(x) } = {x}^{y} }}}$

Answer 2

This is the solution to your problem with Step by Step explanation. Please Mark Me As The Brainliest Answer.

Note : "OR" is given to indicate alternative solution.