I know that the argument of transcendental functions must be dimensionless. I am having difficulty understanding, then, how logarithms can be used in general. In some cases, I have seen that the quantity must be nondimensionalized before a logarithm can be applied. This makes sense to me with single terms on the left- and right-hand sides, for instance, $M/M_o$ could be a non-dimensional mass.
But more generally, how is it possible to take the logarithm of products of variables, e.g., $F = ma$? Here, $\log F = \log m + \log a$, or in dimensional units, $\log MLT^{-2}=\log M+\log LT^{-2}$.
This question is motivated by the formulation of allometry in biology, which is constructed as $y=kx^a$, or in logarithm form, $\log y =\log k + a\log x$. k is taken to be a dimensional constant, but because y and x can have any units, then this violates the nondimensionality of the logarithm. How can this be resolved?
Short answer:
Only take the logarithm of dimensionless quantities:
$$\log\frac{F}{ma}=0,$$
$$\log\frac{y}{kx^a}=0.$$
In the second case, $x$ must itself be a dimensionless parameter, otherwise $x^a$ would be meaningless (unless it is a small integer).
You can also work with reduced quantities, i.e. use reference values that are known to be compatible: using
$$F_0=m_0a_0,$$
$$\log\frac{F}{F_0}=\log\frac{m}{m_0}+\log\frac{a}{a_0},$$
and
$$y_0=k_0x_0^a,$$
$$\log\frac{y}{y_0}=\log\frac{k}{k_0}+a\log\frac{x}{x_0}.$$