Let $$f:{R^ + } \to R$$ , where $${R^ + }$$ is the set of all positive real numbers, be such that $$f\left( x \right) = {\log _e}x$$ Determine Whether $$f\left( {xy} \right) = f\left( x \right) + f\left( y \right)$$ holds.

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Hint: Here a simple function is given, we need to find out whether the equation for the function $f$ holds or not. For that, we have to map the function $f$ from $${R^ + }$$to $$R$$ . The function $f$ is defined, we just need to put $xy$ in the mapping then we can find the required solution. We will get the required result. Formula used: Logarithm formula is defined as, $${\log _a}\left( {bc} \right) = {\log _a}b + {\log _a}c$$Complete step by step answer:It is given that, $$f:{R^ + } \to R$$ , where $${R^ + }$$ is the set of all positive real numbers, be such that$$f\left( x \right) = {\log _e}x$$ We need to find out whether $$f\left( {xy} \right) = f\left( x \right) + f\left( y \right)$$ holds or not.For that, now we are going to put $xy$ in the defined function $f$ we get, $$f\left( {xy} \right) = {\log _e}\left( {xy} \right)$$ Since, we know that, $${\log _a}\left( {bc} \right) = {\log _a}b + {\log _a}c$$ $$ \Rightarrow {\log _e}x + {\log _e}y$$ By using the function $$f\left( {xy} \right) = {\log _e}\left( {xy} \right)$$ $$ \Rightarrow f\left( x \right) + f\left( y \right)$$ Thus, the equation $$f\left( {xy} \right) = f\left( x \right) + f\left( y \right)$$ holds.Hence, the correct answer is option (B).Note: Function: A function $$f:X \to Y$$ is a process or a relation that associates each element x of a set X, the domain of the function to a single element y of another set Y (possibly the same set), the codomain of the function.If the function is called f, this relation is denoted by $$y = f\left( x \right)$$ where the elements x is the argument or input of the function and y is the value of the function or output or the image of x by f.Logarithm: In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.The logarithm of x to base b is denoted as $${\log _b}x$$ .

Let \[f:{R^ + } \to R\] , where \[{R^ + }\] is the set of all positive real numbers, be such that \[f\left( x \right) = {\log _e}x\] Determine Whether \[f\left( {xy} \right) = f\left( x \right) + f\left( y \right)\] holds.