Common and Natural Logarithms - Problem 1

Approximating logarithms without a calculator. For this particular example what I want to do is figure out what the log of 125 is between the term of two integers. To do this first we have to remember that log 125 is the common log, there is no base but we know that’s going to be 10.

This is really log base 10 of 125. What we now need to do is think of what logs we know that lie around 125. We’re trying to think of log base 10 of something that we know. We’re thinking of powers of 10. Log base 100 is something we know. 10 to what power is 100, this is going to be equal to 2.

Then the next number that we need to think about is a number, so this 100 is less than 125. We need to now think of the next number larger than 125 that is also power of 10. That’s going to be 1000. Log of 1000, 10 to what power is equal to 1000, is going to be 3. By that logic we know that the log of 125 has to be bigger than the log of 100 and has to be less than the log of 1000. So the log of 125 has to be between 2 and 3.

Just by inspection you can tell that it’s probably going to be closer to 2 because 125 is significantly closer to 100 than 1000. Just by using some basic common sense sort of thinking it through we can at least figure out logically that this is going to be between 2 and 3.