When I was studying in high school, my teacher taught me about
log(a) + log(b) = log(ab)
log(a) - log(b) = log(a/b)
log(a) / log(b) = log$_b$(a)
Then, I asked my teacher "Why there is no formula log(a) * log(b) = (something) ?".
My teacher didn't know why ?
Now, I still don't know why there is no formula log(a) * log(b) = (something) ?
On
There is currently no well-known function $\;f(x,y)\;$ such that $\;\log(x)\cdot\log(y)=\log(f(x,y)).\;$ That is, the function $\;f(x,y):=x^{\log(y)}=y^{\log(x)}\;$ has not been given a name yet, although it is a valid function. This situation may change at some future time. There are comparatively few named functions but new ones appear sometimes. One such example is the Lambert W function.
For real numbers $x$ and $y$, the equation $(\log x)y = \log (x^y)$ holds.
Thus for real positive numbers $a$ and $b$, from letting $y = \log b$, it follows that $\log(a)\log(b) = \log(a^{\log b})$. Yes one can deduce that $\log a \log b$ is also $\log (b^{\log a})$.
These equations are not mentioned much, perhaps because they can easily be deduced from the other laws (and it doesn't seem all that interesting, at least to the generalist just learning this stuff).