Question

Evaluate:

`log_(√(3) )(729) \\`
Give answer please..
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Answer 1

I’m not sure but
1.66992439152

Answer 2

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

Given Logarithmic expression is

`\rm \longmapsto\: log_( √(3) )(729)`

Let first factorize 729

`\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}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 * 3 * 3 * 3 * 3 * 3}`

`\purple{\rm \longmapsto\:729 = {3}^(6) }`

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

`\purple{\rm \longmapsto\:729 = {( √(3) )}^(12)}`

So,

`\rm \longmapsto\: log_( √(3) )(729)`

can be rewritten as

`\rm \:  =  \: log_( √(3) )( {( √(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_( √(3) )(729) \: = \: 12 \: \: }}}`