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Q.
The value of $k$ for which the equation $\ln ( k \ln x )=\ln x$ has a unique solution, is
Application of Derivatives
Solution:
$\ln ( k \ln x )=\ln x \Rightarrow \ln x =\frac{ x }{ k }$ ....(1)
The line $y =\frac{ x }{ k }$ becomes tangent to the curve $y =\ln x$ at $x = e$ for $k = e$.
So, the solution of the equation (1) is
$x = k \text { at } k = c$