Properties of Log Function \(f(x)=\log_e x\)

Solution

Given:

$$ f(x)=\log_e x $$

where the domain is:

$$ \mathbb{R}^+=\{x:x>0\} $$

(i) Image Set of the Domain of \(f\)

The logarithmic function \(f(x)=\log_e x\) can take every real value.

Therefore, the image set (range) is:

$$ \mathbb{R} $$

(ii) Find \(\{x : f(x)=-2\}\)

We have:

$$ \log_e x=-2 $$

Converting into exponential form:

$$ x=e^{-2} $$

Hence,

$$ \{x:f(x)=-2\}=\{e^{-2}\} $$

(iii) Verify whether \(f(xy)=f(x)+f(y)\)

We know:

$$ f(xy)=\log_e(xy) $$

Using the logarithmic property:

$$ \log_e(xy)=\log_e x+\log_e y $$

Therefore,

$$ f(xy)=f(x)+f(y) $$

Hence, the given property holds true for all positive real numbers \(x\) and \(y\).