Ratio of expected values of squares of errors in power quantities, in dB
Your question raises many very present concerns about measuring differences, and optimizing, in signal/image processing. Measuring can help:


to compare different outcomes from two different processes, 
to optimize processing per se (for denoising, restoration).


Most use cost functions related to  $p$-norms (when $p\ge 1$) or quasi-norms (when $0&lt;p\le 1$): $$\mathcal{C}_p(a,\hat{a})=\left(\sum_k|a_k-\hat{a}_k|^p\right)^{1/p}\,.$$ Those functions are symmetric, and are consistent with the physical units of $a$. The behavior of $\mathcal{C}_p$ depends on power $p$: classical norm inequalities tells you that $\mathcal{C}_p(a,\hat{a})\ge \mathcal{C}_q(a,\hat{a})$ when $p\le q$. Those (quasi-)norms are $1$-homogeneous. Using different $p$ or $q$ can be useful, especially in a sparsity context, see for instance references in: Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed \ell_1/\ell_2 Regularization. Here is a rendition of classical penalties: 

Thus, quasi-norm or norm ratios are used to compare estimates, in a unitless fashion. The $\ell_p$ signal-to-noise ratios are typically of the shape: 

$$\frac{\mathcal{C}_p(a,0)}{\mathcal{C}_p(a,\hat{a})}\,.$$

They are unit-less (ratios of same units). To better express orders of magnitude, one usually take logarithms. The logarithms can also been interpreted as unit-less measure. Interestingly, either $\log$ or $\ell_p/\ell_q$ ratios (like $\ell_1/\ell_2$) have been used a non-convex penalties in signal restoration. 

Your functions with power $\alpha$ can be written as:

$$B\log_b \frac{\mathcal{C}_p(a^\alpha,\hat{a}_2^\alpha)}{\mathcal{C}_p(a^\alpha,\hat{a}_1^\alpha)}\,.$$

Now, those logarithms of ratios are unit-less, and are usually employed in a relative way, for comparison only. With $B&gt;0$, $b&gt;1$, $t \to B \log_b (t)$ is a monotonic increasing function. So, for value comparison only, we don't really care about the values for $b$ and $B$. However, for consistency in dB scale (as Olli Niemitalo already answered properly), signal-to-noise ratios tend to be computed in $10\log_{10}$ of energy measures, so homogeneously:


$ 20\log_{10}(|a|)$
$ 10\log_{10}(|a|^2)$
$ 5\log_{10}(|a|^4)$


make sense. However, it is not easy to compare them objectively, but they can be plotted on similar graph.