TITLE: The logarithm is non-linear! Or isn't it?
QUESTION [39 upvotes]: The logarithm is non-linear
Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do something like this:

We have
  $$
\ln(x + y) \neq \ln(x) + \ln(y) \quad\text{and}\quad \ln(\lambda \cdot x) = \ln(\lambda) + \ln(x) \neq \lambda \cdot \ln(x),
$$
  and therefore $\ln$ is not linear.

And indeed, the literature is abundant with the claim that...

... a function $f : V \to W$ is linear, if and only if
  $$
f(x + y) = f(x) + f(y) \quad\text{and}\quad f(\lambda \cdot x) = \lambda \cdot f(x)
$$
  for all $x,y$ and all scalars $\lambda$.

Often, there is no hint that the symbols $+$ and $\cdot$ on the left belong to $V$, whereas the symbols $+$ and the $\cdot$ on the right belong to $W$.
The logarithm is linear
My proof that the logarithm is a linear function goes like this:

$$\ln(x \cdot y) = \ln(x) + \ln(y) \quad\text{and}\quad \ln(x^\lambda) = \lambda \cdot \ln(x).$$

The rationale for this is that $\ln : \mathbb{R}_{>0} \to \mathbb{R}$, i.e., the logarithm is a function from the $\mathbb{R}$-vector space $\mathbb{R}_{>0}$ (the positive-real numbers), to the $\mathbb{R}$-vector space $\mathbb{R}$ (the real numbers). Vector addition in $\mathbb{R}_{>0}$ is, however, not usual addition, but multiplication. Likewise, scalar multiplication in $\mathbb{R}_{>0}$ is not usual multiplication, but potentiation.
In fact, the linear-algebra definition of linearity is (e.g. Ricardo, 2009; Bowen and Wang, 1976):

A function $f : V \to W$ from a vectors space $(V,\oplus,\odot)$ over a field $F$ to a vector space $(W,\boxplus,\boxdot)$ over $F$ is linear if and only if it satisfies
  $$
f(x \oplus y) = f(x) \boxplus f(y) \quad\text{and}\quad f(\lambda \odot x) = \lambda \boxdot f(x)
$$
  for all $x,y \in V$ and $\lambda \in F$.

Another proof goes as follows:

The logarithm is an isomorphism between the vector space of positive-real numbers to the vector space of real numbers. And as every isomorphism is a linear function, so is the logarithm.

Question
We have two conflicting statements here:

The logarithm is non-linear.
The logarithm is linear.

Can both statements be correct simultaneously, depending on something I cannot imagine now? But wouldn't this also imply that two conflicting concepts of linearity exist?
Or is this a case of sloppy notation, e.g., abuse of the same symbol $+$ for vector addition or $\cdot$ for scalar multiplication even though two different vector spaces are involved?
Update
The solutions given to rescue the first statement haven't convinced me yet, because they are inconsistent:

Using usual addition and multiplication on $\mathbb{R}_{>0}$ implies that $(\mathbb{R}_{>0},+,\cdot)$ is not a vector space anymore. But a precondition of the linearity proof is that the domain and the range of $f$ are vector spaces.
Allowing the domain of $\ln$ to be $\mathbb{R}$ with usual addition and multiplication instead of $\mathbb{R}_{>0}$ doesn't work, because then the image of $\ln$ is the set of complex numbers.
A mathematically consistent definition of "linearity" for subsets (but not subspaces) of a vector space was given in a comment by @Alex G. Let $S$ be an arbitrary subset of a real vector space $V$, and let $W$ be a real vector space. A function $f : S \to W$ is called "linear" if for all $x,y \in S$ such that $x+y \in S$, then $f(x+y) = f(x)+f(y)$, and for all $x \in S$, $k \in \mathbb{R}$ such that $kx \in S$, then $f(kx)=k⋅f(x)$. However, this definition is not what is meant by the concept of linearity coming from linear algebra. One would actually need to use another term for "linearity" here.

REPLY [2 votes]: The error in the question is to assume that people say "linear" only in the context of algebra (where log can be interpreted as linear in the way you explained) when in fact the usage almost everywhere else is incompatible with this alternative interpretation.  Examples include

linear functions in geometry (and physics, science, economics, etc), as ones whose graph belongs to a straight line
linear growth and linear estimates in analysis
linear time and space in computational complexity theory
sublinear, superlinear, quasilinear

Even in algebra there are

linear algebraic groups, general linear group $GL_n$, special linear group $SL_n$.  There is no linear algebraic group representation using the logarithm.

In some of these uses there is no $\mathbb{R}$ or $\mathbb{R}^{\ast}$ involved.
It would be unbelievable to argue that in all these cases there is some hidden choice of "linearizing structure" that was not specified but can be consistent with log(x) being linear, so that LOGSPACE is really linear space for some other convention than the one actually in use in this universe.
Thus the resolution of this "paradox" is to recognize that logarithm is necessarily nonlinear in most situations, and in algebra having it be nonlinear is just the default option that one can over-ride locally (with suitable warning to the reader or listener) in a particular context.