Explain how the common logarithm of a number can tell you how many digits the number has.

Question

I don’t understand this question. I didn’t know you could tell the digits of a common logarithmic number!!!

Answer 1

It's not that it tells you the digits of a logarithmic functions, logarithm can be used to tell you the digits of whole numbers.

For example, the number 12345 has five digits.

Anything with 5 digits is going to fall between the numbers 10000 and 100000, not including the last number.

The log of 10000 is 4 because 10⁴ = 10000

The log of 100000 is 5 because 10⁵ = 100000

Therefore if we take the log of any number in between 10000 and 100000, we should get something in between 4 and 5.

For example log(12345) equals 4.091

log(98765) = 4.995

We see that with 5 digit numbers if we take the log we always get something that starts with 4

This pattern continues. With 6 digit numbers, the log starts with 5. With 7 digit numbers, the log starts with 6 and so on.