Question

pls give solution #log #class 11 # mathematics # IIT-JEE​

Answer 1

$\mathsf{Given :\;60^a = 3}$

$\bigstar\;\;\textsf{We know that : \large\boxed{\mathsf{When\;b^y = x \implies \log_b(x) = y}}}$

$\mathsf{\implies a = \log_{60}(3)}$

$\mathsf{Given :\;60^b = 5}$

$\mathsf{\implies b = \log_{60}(5)}$

$\mathsf{Question\;is:\;12^{\dfrac{1 - a - b}{2(1 - b)}}}$

$\mathsf{\implies 12^{\dfrac{1 - (a + b)}{2(1 - b)}}}$

Substituting the values of a and b, We get :

$\mathsf{\implies 12^{\dfrac{1 - [\log_{60}(3) + \log_{60}(5)]}{2[1 - \log_{60}(5)]}}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{\log_b(x) + \log_b(y) = \log_b(x.y)}}}}$

$\mathsf{\implies 12^{\dfrac{1 - (\log_{60}(3\times 5)}{2[1 - \log_{60}(5)]}}}$

$\mathsf{\implies 12^{\dfrac{1 - \log_{60}(15)}{2[1 - \log_{60}(5)]}}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{\log_b(b) = 1}}}}$

$\implies \mathsf{1\;can\;be\;written\;as\;\log_{60}(60)}$

$\mathsf{\implies 12^{\dfrac{\log_{60}(60) - \log_{60}(15)}{2[\log_{60}(60) - \log_{60}(5)]}}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{\log_b(x) - \log_b(y) = \log_b\bigg(\dfrac{x}{y}\bigg)}}}}$

$\mathsf{\implies 12^{\dfrac{\log_{60}\bigg(\dfrac{60}{15}\bigg)}{2\log_{60}\bigg(\dfrac{60}{5}\bigg)}}}$

$\mathsf{\implies 12^{\dfrac{\log_{60}(4)}{2\log_{60}(12)}}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{\dfrac{\log_c(x)}{\log_c(y)} = log_y(x)}}}}$

$\mathsf{\implies 12^{\dfrac{\log_{12}(4)}{2}}}$

$\mathsf{\implies 12^{\dfrac{\log_{12}(2^2)}{2}}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{\log_b(x^y) = y.\log_b(x)}}}}$

$\mathsf{\implies 12^{\dfrac{2\log_{12}(2)}{2}}}$

$\mathsf{\implies 12^{\log_{12}(2)}}$

$\bigstar\;\;\textsf{We know that : \large{\boxed{\mathsf{b^{\log_b(x)} = x}}}}$

$\implies \large\boxed{\mathsf{12^{\log_{12}(2)} = 2}}$