Logarithmic base $10$ relative and absolute error

Question

I have a question about what does $\log10$ error in absolute and relative sense represent. Let $y$ be true value and $\hat{y}$ by approximation value of $y$. Then:

1. Absolute (Forward) Error: $\|\hat{y}-y\|$
2. Relative (Forward) Error: $\frac{\|\hat{y}-y\|}{\|y\|}$

So am I right in saying that:

1. $ - \log_{10}(\|\hat{y}-y\|)$ measures the number of digits to which $\hat{y}$ is accurate to $y$.
2. $ - \log_{10}(\frac{\|\hat{y}-y\|}{\|y\|})$ measures the number of decimal points to which $\hat{y}$ is accurate to $y$.

The reason I say the latter is number of decimals is because $$ - \log_{10}(\frac{\|\hat{y}-y\|}{\|y\|}) = \log_{10}(\|y\|) - \log_{10}(\|\hat{y}-y\|)$$ so you get rid off the magnitude of $y$ which is above the decimal point.

Am I right in making this assumption? I am aware of the fact that above quantities are not integers, so we need to round down.