[Community Question] Calculus: Logaritmic functions in terms of vector space theory

One of our user asked:

We can consider $\mathbb{R}^+$ with two operation:

+: $\mathbb{R}^+ \times \mathbb{R}^+\to \mathbb{R}^+$

that maps $(a,b)$ to $a+b:=ab$ and

$ \cdot : \mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}^+$

that maps $(\lambda,a)$ to $a^\lambda$.

With respect to these operation we have that $\mathbb{R}^+$ is a vector space.

We want find an example of linear application from $\mathbb{R}$ to $\mathbb{R}^+$. We can fix $a\in \mathbb{R}^+$ and we can define

$f_a:\mathbb{R}\to \mathbb{R}^+$ such that maps every $\lambda$ to $ f_a(\lambda):=a^\lambda$.

From the rules of powers, the map $f_a$ is a linear map for every $a\in \mathbb{R}^+$.

A natural question can be if all linear maps from $\mathbb{R}$ to $\mathbb{R}^+$ are an exponential maps.

The answer is positive because for each linear map $F: \mathbb{R}\to \mathbb{R}^+$ we have that

$F(\lambda)=F(\lambda \cdot 1)=(F(1))^\lambda=f_{F(1)}(\lambda)$

so

$F=f_{F(1)}$

We can also observe that the map $f_a$ is bijective and its inverse is the logaritmic function $log_a \mathbb{R}^+\to \mathbb{R}$ that it is linear because of logartmic rules or because it is the inverse of a linear map.

So the logaritmic maps are elements of the dual of $\mathbb{R}^+$.

A natural question can be if all non-zero functional of $\mathbb{R}^+$ must be a logaritmic function.

The answer is positive because $G\in (\mathbb{R}^+)^*/\{0\}$ is a linear map between two vector space of dimension 1 so it is bijective but $G^{-1}: \mathbb{R}\to \mathbb{R}^+$ is a linear map so $G^{-1}=f_{G^{-1}(1)}$.

Hence $G=(f_{G^{-1}(1)})^{-1}=log_{G^{-1}(1)}$.

So the dual of $\mathbb{R}^+$ is

$ (\mathbb{R}^+)^*=\{log_a : a\in \mathbb{R}^+\}$

It is correct?