[tex] \large\underline{\sf{Solution-}}[/tex]
Given Logarithmic expression is
[tex]\rm \longmapsto\: log_{ \sqrt{3} }(729) [/tex]
Let first factorize 729
[tex]\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}[/tex]
So,
[tex] \purple{\rm \longmapsto\:729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3}[/tex]
[tex] \purple{\rm \longmapsto\:729 = {3}^{6} }[/tex]
[tex] \purple{\rm \longmapsto\:729 = {( [{ \sqrt{3} ]}^{2}) }^{6} }[/tex]
[tex] \purple{\rm \longmapsto\:729 = {( \sqrt{3} )}^{12}}[/tex]
So,
[tex]\rm \longmapsto\: log_{ \sqrt{3} }(729) [/tex]
can be rewritten as
[tex]\rm \: = \: log_{ \sqrt{3} }( {( \sqrt{3}) }^{12} ) [/tex]
We know,
[tex] \purple{\rm \longmapsto\:\boxed{\tt{ log_{a}( {a}^{x} ) \: = \: x \: }}}[/tex]
So, using this identity, we get
[tex]\rm \: = \: 12[/tex]
Hence,
[tex] \green{\rm\implies \:\boxed{\tt{ \: \: log_{ \sqrt{3} }(729) \: = \: 12 \: \: }}}[/tex]
