Log Basics.

According to the laws of Logarithms ,

$log_a {b}^{c}= c log_a b$

Now let $log {3}^{1000}= log x$

$100 \times {log3}=log x$

log x = 477.1

In any logarithm, The number BEFORE the decimal point denotes the characteristic, and AFTER the decimal point, It denotes Mantissa. You can read more about it here

The characteristic of a number is always 1 less than the number of digits in the number.

Here, The Characteristic of x is 477.

Hence, The number of digits is 478.

Therefore, the answer is $\boxed {478}$