Added: the first inequality is not tied to $\log 2$ well enough. The second one, from Robin's dissertation, does give
$$ \frac{\log d(n)}{\log 2} \leq \left( \frac{\log n}{\log \log n} \right) \left( 1 + \frac{1.934850967971...}{\log \log n} \right)$$
With equality at $n = 6983776800 = 2^5 \cdot 3^3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19$ and $d(n) = 2304,$
$$ d(n) \leq n^{ \left( \frac{\log 2}{\log \log n} \right) \left( 1.5379398606751... \right)} = n^{ \left( \frac{1.0660186782977...}{\log \log n} \right) }. $$ Full details of the proof appear in J.-L. Nicolas et G. Robin. Majorations explicites pour le nombre de diviseurs de n, Canad. Math. Bull., 26, 1983, 485--492. The next two appear in the dissertation of Robin.
With equality at a number $n$ near $6.929 \cdot 10^{40},$
$$ d(n) \leq n^{ \left( \frac{\log 2}{\log \log n} \right) \left( 1 + \frac{1.934850967971...}{\log \log n} \right)}. $$ Compare this one with Theorem 317 in Hardy and Wright, attributed to Wigert (1907),
$$ \limsup \frac{\log d(n) \log \log n}{\log n} = \log 2. $$
With equality at a number $n$ near $3.309 \cdot 10^{135},$
$$ d(n) \leq n^{ \left( \frac{\log 2}{\log \log n} \right) \left( 1 + \frac{1}{\log \log n} + \frac{4.762350121177...}{\left(\log \log n \right)^2} \right)} $$