Question

Log 3 base x × log 3 base x/3 + log 3 base x/81

Answer 1

SOLUTION

TO DETERMINE

The value of x when

$\displaystyle \sf{ log_{x}(3). log_{ \frac{x}{81} }(3) = log_{ \frac{x}{729} }(3) }$

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

$\displaystyle \sf{ log_{x}(3). log_{ \frac{x}{81} }(3) = log_{ \frac{x}{729} }(3) }$

We now solve for x as below

$\displaystyle \sf{ log_{x}(3). log_{ \frac{x}{81} }(3) = log_{ \frac{x}{729} }(3) }$

$\displaystyle \sf{ \implies \: \frac{1}{ log_{3}(x)} . \frac{1}{log_{3}( \frac{x}{81} )} = \frac{1}{log_{3}( \frac{x}{729} )} }$

$\displaystyle \sf{ \implies \: log_{3}(x). log_{3} \bigg( \frac{x}{81} \bigg) = log_{3} \bigg( \frac{x}{729} \bigg) }$

$\displaystyle \sf{ \implies \: log_{3}(x). \bigg( log_{3}(x) - log_{3}(81) \bigg) = \bigg( log_{3}(x) - log_{3}(729) \bigg) }$

$\displaystyle \sf{ \implies \: log_{3}(x). \bigg( log_{3}(x) - log_{3}( {3}^{4} ) \bigg) = \bigg( log_{3}(x) - log_{3}( {3}^{6} ) \bigg) }$

$\displaystyle \sf{ \implies \: log_{3}(x). \bigg( log_{3}(x) - 4log_{3}( {3}^{} ) \bigg) = \bigg( log_{3}(x) - 6 log_{3}( {3}^{} ) \bigg) }$

$\displaystyle \sf{ \implies \: log_{3}(x). \bigg( log_{3}(x) - 4 \bigg) = \bigg( log_{3}(x) - 6 \bigg) }$

Let

$\displaystyle \sf{ y = log_{3}(x) }$

Then above equation becomes

$\displaystyle \sf{ y(y - 4) = y - 6 }$

$\displaystyle \sf{ \implies \: {y}^{2} - 5y + 6 = 0 }$

$\displaystyle \sf{ \implies \: y = 2 \: \: \: and \: \: \: 3 }$

Now y = 2 gives

$\displaystyle \sf{ log_{3}(x) = 2 }$

$\displaystyle \sf{ \implies \: x = {3}^{2} = 9 }$

y = 3 gives

$\displaystyle \sf{ log_{3}(x) =3 }$

$\displaystyle \sf{ \implies \: x = {3}^{3} = 27 }$

FINAL ANSWER

Hence the required solution is x = 9 , 27

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Answer 2

SOLUTION

TO DETERMINE

The value of x when

FORMULA TO BE IMPLEMENTED

We are aware of the formula on logarithm that

EVALUATION

Here the given equation is

We now solve for x as below

Let

Then above equation becomes

Now y = 2 gives

y = 3 gives

FINAL ANSWER

Hence the required solution is x = 9 , 27