Given that 0.475 < log base 10 of 3 < 0.478, how
many digits are in the number 350?
We have to get 350 into the form 10x
We need to find x such that
350 = 10x
Take logs (base 10 understood) of both sides
log(350) = log(10x)
Using rules of logarithms:
50·log(3) = x
Solve for log(3)
log(3) = x/50
We are given that
0.475 < log(3) < 0.478
So substitute x/50 for log(3)
0.475 < x/50 < 0.478
Multiply thru by 50
23.75 < x < 23.9
Raise 10 to all three powers:
1023.75 < 10x < 1023.9
1023.75 < 350 < 1023.9
Integers equal to or greater than 100 but less than 101
have 1 digit.
Integers equal to or greater than 101 but less than 102
have 2 digits.
Integers equal to or greater than 102 but less than 103
have 3 digits.
Integers equal to or greater than 103 but less than 104
have 4 digits.
... ... ... ... ... ... ... ...
Integers equal to or greater than 1023 but less than 1024
have 24 digits.
1023 < 1023.75 < 350 < 1023.9 < 1024
Thus 350 has 24 digits.
Checking with the calculator: 350 = 7.176879877×1023
Edwin
AnlytcPhil@aol.com