Without log table!

We need to solve the equation:

$log_{e}2 \times log_{b}625=log_{10}16 \times log_{e}10$

$log_{b}625=log_{10}16 \times \frac{log_{e}10}{log_{e}2}$

$log_{b}625=log_{10}16 \times log_{2}10$

$log_{b}625=log_{10}16 \times \frac{1}{log_{10}2}$

$log_{b}625=\frac{log_{10}16}{log_{10}2}$

$log_{b}625=log_{2}16$

$log_{b}625=log_{2}2^{4}$

$log_{b}625=4log_{2}2$

$log_{b}625=4\times1$

$log_{b}625=4$

So, we get the equation as: $b^{4}=625$

$b^{4}=5^{4}$

$b=5$

Thus, the answer is: $b=\boxed{5}$

Here, number inside bracket represents the base.

log (b) 625 = log 16 x log (e) 10 / log (e) 2

log (b) 625 = log 16 x log (2) 10

log (b) 625 = log 16 x [1 / log (10) 2]

log (b) 625 = log 16 / log 2

log (b) 625 = log (2) 16

log (b) 625 = 4

b = 5