Count the digits

$125^{100}=5^{300}=\frac{10^{300}}{2^{300}}$

Also, $2^{300}=1024^{30}=10^{90}\times 1.024^{30}$

Therefore, $125^{100}=\frac{10^{210}}{1.024^{30}}$

By estimation with the binomial expansion, we can estimate that $1<1.024^{30}<10$

Therefore, $125^{100}$ has 210 digits.

The number of digits of a number raised to some power is

for e.g in $a^{x}$ In base 10

$\left\lfloor xloga \right\rfloor +1$

So the answer comes out to be

$\left\lfloor 100log125 \right\rfloor +1$ = $210$