Question

find the number of digits in the number:
7^600 * 3^8000

can someone show me how to solve this?

Answer 1

Is this number theory?

Answer 2

no just an algebra class

Answer 3

I got 4325. What did you get?

Answer 4

I also got 4325!

Answer 5

how did you get that? that is correct

Answer 6

It's simple:\[7^{600}\to\left \lceil 600\log7 \right \rceil=508\]and\[3^{8000}\to\left \lceil 8000\log3 \right \rceil=3817.\]Therefore,\[508+3817=4325.\]

Answer 7

If you had a number like 10^2 you know it has 3 digits (it is 100)

so change the number to a power of 10
\[10^{600\log_{10}7}10^{8000 \log_{10}3}= 10^{600\log_{10}7+8000 \log_{10}3}\]
the exponent is 4324.0288862..
or
\[ 10^{0.028862} 10^{4324} = 1.0687 \cdot 10^{4324}\]
so the number of digits in the number is 4325