BASES AND LOGARITHMS

Question

When we work in Base 5, for example, we're told that the only digits we can use are 0,1, 2, 3 & 4. So, for example, the first eight positive whole numbers in Base 5 are 1, 2, 3, 4, 10, 11, 12, 13, ... However, when we work in Base 5 with regards to logarithms, we're able to use all the standard (Base 10) digits at our disposal. That is, we can write log₅ 625 = 4. Why are we allowed to use the digit '6' when we're working in base 5?

Answer 1

John,

we use the word “base” with two different meanings.

(1) in base 5 representation, the only digits used are 0, 1, 2, 3, and 4, as you said. Further, the rightmost digit is the units digit, second right-most digit is the 5’s digit, the next to the left is the 5² digit, and so on.

(2) the “logarithm to the base 5 of x” means the number value y such that 5^y = x. This time, we are not discussing a particular representation, but rather the exponent of 5.

Does this help?