Question

$\huge{\fbox{\fbox{\orange{\mathfrak{Explanation\:Required}}}}}$

Solve the above equation.​

Answer 1

Solution :

$\mathrm{ log_{1/3}9 \sqrt{3} }$

It can be written as

$\mathrm{ = log_{1/( \sqrt{3} ) ^{2} }3^{2} \sqrt{3} }$

$\mathrm{ = log_{1/( \sqrt{3} ) ^{2} } \big( (\sqrt{3}) ^{2} \big) ^{2} \sqrt{3} }$

Using ( a^m )^n = a^( mn ) we get,

$\mathrm{ = log_{1/( \sqrt{3} ) ^{2} } (\sqrt{3}) ^{4} \sqrt{3} }$

Using a^m × a^n = a^( m + n ) we get,

$\mathrm{ = log_{1/( \sqrt{3} ) ^{2} } (\sqrt{3}) ^{5} }$

Using 1 / a^m = a^( - m ) we get,

$\mathrm{ = log_{( \sqrt{3} ) ^{ - 2} } (\sqrt{3}) ^{5} }$

$\text{Using } \sf log_{a^m }n = \dfrac{1}{m} log_{a}n$

$\mathrm{ = - \dfrac{1}{2} log_{ \sqrt{3} } (\sqrt{3}) ^{5} }$

Using Power rule - log a^m = m.log a we get,

$\mathrm{ = - \dfrac{1}{2} ( 5 \times log_{ \sqrt{3} } \sqrt{3}})$

$\mathrm{ = - \dfrac{5}{2} log_{ \sqrt{3} } \sqrt{3}}$

Using log_a a = 1 we get,

$\mathrm{ = - \dfrac{5}{2} \times 1}$

$\mathrm{ = - \dfrac{5}{2}}$

Therefore the value of the given expression is - 5 / 2.

Answer 2

Answer:

$\large\boxed{\sf{-\dfrac{5}{2}}}$

Step-by-step explanation:

To find the value of,

$log_{ \frac{1}{3} }(9 \sqrt{3} )$

Let the required value be 'x'

Therefore, we have the equation,

$= > log_{ \frac{1}{3} }(9 \sqrt{3} ) = x$

But, we know that,

If,

• $log_{x}(y) = m$

Then,

• $y = {x}^{m}$

Therefore, we will get,

$= > {( \dfrac{1}{3}) }^{x} = 9 \sqrt{3}$

But, we know that,

• ${y}^{ - m} = {( \dfrac{1}{y} )}^{m}$

Therefore, we will get,

$= > {3}^{ - x} = 9 \sqrt{3} \\ \\ = > {3}^{ - x} = {3}^{2} \times {3}^{ \frac{1}{2} }$

But, we know that,

• ${a}^{m} \times {a}^{n} = {a}^{m + n}$

Therefore, we will get,

$= > {3}^{ - x} = {3}^{(2 + \frac{1}{2}) } \\ \\ = > {3}^{ - x} = {3}^{ \frac{5}{2} }$

Now, comparing the both sides, we observed that bases are same i.e., 3

Therefore, powers / exponent will also be same

Therefore, we will get,

$= > - x = \dfrac{5}{2} \\ \\ = > x = - \dfrac{5}{2}$

Hence, the required value is -5/2.