Question

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.

Answer 1

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 ${\log _b}x$ .