How to find no. of digits of a large product

Question

The question is:

The product of 45,454,545,454,545 and 1,234 contains how many digits?

I dont know how to solve it other than typing it in my calculator, but that method is wrong too.

Answer 1

Hint: Use scientific notation. (i.e. write $45,454,545,454,545 =4.5454545454545*10^{13}$)

Answer 2

The product is greater than $(4.0\times 10^{13})\times 10^3=4.0\times 10^{16}$ and less than $(5.0\times 10^{13})(2.0\times 10^3)=1.0\times 10^{17}$, hence it has $17$ digits.