Question

no. of solutions of sq. root log3(3x^2).log9 (81x) = log9(x^3) is-​

Answer 1

SOLUTION

TO DETERMINE

The number of solutions

$\displaystyle \sf{ \sqrt{ log_{3}(3 {x}^{2} ) . log_{9}(81x) } = log_{9}( {x}^{3} ) }$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on logarithm that

$\sf{1. \: \: \: log( {a}^{n} ) = n log(a) }$

$\sf{2. \: \: log(ab) = log(a) + log(b) }$

$\displaystyle \sf{3. \: \: log \bigg( \frac{a}{b} \bigg) = log(a) - log(b) }$

$\sf{4. \: \: log_{a}(a) = 1}$

EVALUATION

Here the given equation is

$\displaystyle \sf{ \sqrt{ log_{3}(3 {x}^{2} ) . log_{9}(81x) } = log_{9}( {x}^{3} ) }$

We now solve for x as below

$\displaystyle \sf{ \sqrt{ log_{3}(3 {x}^{2} ) . log_{9}(81x) } = log_{9}( {x}^{3} ) }$

$\displaystyle \sf{ \implies \: \sqrt{ \: [ \: log_{3}(3 ) +log_{3}( {x}^{2} ) ] . [ \: log_{9}(81) +log_{9}(x) ] } = log_{9}( {x}^{3} ) }$

$\displaystyle \sf{ \implies \: \sqrt{ \: [ \: 1 +2log_{3}( x ) ] . [ \: 2 +log_{9}(x) ] } = 3log_{9}( x) }$

$\displaystyle \sf{ \implies \: \sqrt{ \: [ \: 1 +4log_{9}( x ) ] . [ \: 2 +log_{9}(x) ] } = 3log_{9}( x) }$

Let

$\displaystyle \sf{y = log _{9}( x ) }$

Then above equation becomes

$\displaystyle \sf{ \implies \: \sqrt{(1 + 4y)(2 + y)} = 3y }$

$\displaystyle \sf{ \implies \: (1 + 4y)(2 + y) = 9 {y}^{2} }$

$\displaystyle \sf{ \implies \: 2 + y + 8y + 4 {y}^{2} = 9 {y}^{2} }$

$\displaystyle \sf{ \implies \: 5 {y}^{2} - 9y - 2 = 0 }$

$\displaystyle \sf{ \implies \: 5 {y}^{2} - 10y + y - 2 = 0 }$

$\displaystyle \sf{ \implies \: (y - 2)(5y + 1) = 0 }$

$\displaystyle \sf{ \implies \: y = 2 \: ,\: - \frac{1}{5} }$

Now

$\displaystyle \sf{ \: y = 2 \: \: gives}$

$\displaystyle \sf{ log _{9}( x ) = 2 }$

$\displaystyle \sf{ \implies \: x = {9}^{2} }$

$\displaystyle \sf{ \implies \: x =81 }$

Now

$\displaystyle \sf{ \: y = - \frac{1}{5} }$ does not satisfy the given equation

So the only solution is x = 81

FINAL ANSWER

The number of solutions = 1

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