Question

[MULTIPLE ANSWERS CAN BE CORRECT ]
​

Answer 1

$x = {13}^{ log_{k}(7) }$

$\implies log_{7}( {13}^{ log_{k}7} ) + log_{13} ({13}^{ log_{k}7 } ) = 1$

$\blue{ \boxed{log( {a}^{m} ) = m. log(a) }}$

$\implies { log_{k}7}(log_{7}{13}) + { log_{k}7}( log_{13}13) = 1$

$\implies log_{k}7( log_{7}13 + log_{13}13 ) = 1$

$\blue{ \boxed{ log_{a}(a) = 1}}$

$\implies log_{7}(13) + 1 = \frac{1}{ log_{k}(7) }$

$\blue{ \boxed{ \frac{1}{ log_{b}(a) }= log_{a}(b) }}$

$\orange{\therefore k = 91}$

$\implies log_{7}(13) + 1 = log_{7}(k)$

$\implies log_{7}(13) - log_{7}(k) = - 1$

$\blue{ \boxed{ log(a) - log(b) = log(\frac{a}{b} ) }}$

$\implies log_{7}( \frac{13}{k} ) = - 1$

$\tt \: if \: \blue{ \boxed{ log_{a}(N) = x }} \: \tt then :$

$\blue{ \boxed{a^x=N}}$

$\implies {7}^{ - 1} = \frac{13}{k}$

$\implies \frac{1}{7} = \frac{13}{k}$

$\implies k = 7 \times 13$

$\orange{ \therefore k = 91}$