Question

if
$log_{a}(243 \times \sqrt[5]{9} = 2.7$
Then find the value of a.​

Answer 1

$\underline{\huge{Answer:-}}$

Step-by-step explanation:

We are given that

$\implies \: \large \mathtt{ log_{a}(243 \times \sqrt[5]{9} = 2.7 } \\ \\ \ \: \: \: \: \: \: \: \: \: \: \large \mathtt{ log_{a}( {3}^{5} \times ( {3}^{2}) {}^{ \frac{1}{5} } = \frac{27}{10} }$

$\implies \: \large \mathtt{ log_{a}( {3}^{5} \times {3}^{ \frac{2}{5} }) = \frac{27}{10} }$

$\implies \: \large \mathtt{ log_{a}( {3}^{5 + \frac{2}{5} }) = \frac{27}{10} }$

$\implies \: \large \mathtt{ log_{a}( {3}^{ \frac{27}{5} }) = \frac{27}{10} }$

$\implies \: \large \mathtt{ \frac{27}{5} log_{a}3 = \frac{27}{10} }$

$\implies \: \large \mathtt{ log_{a}(3) = \frac{27}{10} \times \frac{5}{27} }$

$\implies \: \large \mathtt{ log_{a}(3) = \frac{1}{2} }$

$\implies \: \large \mathtt{ {a}^{ \frac{1}{2} } = 3}$

Now square both sides

$\implies \: \large \mathtt{a = {3}^{2} = 9 }$

Answer 2

Answer:

$\underline{\huge{Answer:-}}$

We are given that

$\begin{gathered} \implies \: \large \mathtt{ log_{a}(243 \times \sqrt[5]{9} = 2.7 } \\ \\ \ \: \: \: \: \: \: \: \: \: \: \large \mathtt{ log_{a}( {3}^{5} \times ( {3}^{2}) {}^{ \frac{1}{5} } = \frac{27}{10} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ log_{a}( {3}^{5} \times {3}^{ \frac{2}{5} }) = \frac{27}{10} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ log_{a}( {3}^{5 + \frac{2}{5} }) = \frac{27}{10} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ log_{a}( {3}^{ \frac{27}{5} }) = \frac{27}{10} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ \frac{27}{5} log_{a}3 = \frac{27}{10} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ log_{a}(3) = \frac{27}{10} \times \frac{5}{27} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ log_{a}(3) = \frac{1}{2} } \\ \end{gathered}$

$\begin{gathered} \implies \: \large \mathtt{ {a}^{ \frac{1}{2} } = 3} \\ \end{gathered}$

Now square both sides

$\begin{gathered} \implies \: \large \mathtt{a = {3}^{2} = 9 } \\ \end{gathered}$