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Feynman, in his Lectures on Physics, points out that the logarithm, while introduced late in the modern curriculum, is just as mathematically fundamental as roots, if not division. Subtraction is the inverse of addition, division of multiplication. Taking powers isn't symmetric -- 2^3 <> 3^2 -- so we get two inverses, root and logarithm. Given a^b = c, you can ask for a = c^(1/b) or for b = log_a c.
And, I realized while lying in bed this orning, for natural numbers the logarithm is actually much easier to calculate. Subtraction (a-b) can be thought of as subtracting 1 b times. Division (a/b) can be thought of as how many times you can subtract b from a until hitting 0 -- no guesswork needed (that's for long division.) And for log_a c (log, base a, of c) just count how many times you can divide c by a until reaching 1. (If you don't reach 0 or 1, of course, the division or log doesn't exist in the natural numbers.) Finding the roots of natural numbers takes guesswork or search.
In the real numbers it's reversed; there's convenient algorithms for finding roots, which you can find and verify without needing calculus (though calculus gives a proof), while finding logs needs tables of pre-calculated values.
UPDATE: Well, findings logs is a generalization of the natnum procedure; instead of dividing by your base, you divide by powers of your base. E.g in finding log_10 (2), you divide 2 by the fourth root of 10 (the square root is too big), then by the 32nd root of 10, and so on, and the log is the sum of the root powers (1/4 + 1/32 +...) You need the table of the various roots of 10.
Or, of course, you could just do trial and error. "Third root of 10 is bigger than 2, fourth root is smaller, so the log must be a rational in between..." It'll help to use the root algorithms, rather than needing trial and error on those, but it'd still be a fair bit of work.
And, I realized while lying in bed this orning, for natural numbers the logarithm is actually much easier to calculate. Subtraction (a-b) can be thought of as subtracting 1 b times. Division (a/b) can be thought of as how many times you can subtract b from a until hitting 0 -- no guesswork needed (that's for long division.) And for log_a c (log, base a, of c) just count how many times you can divide c by a until reaching 1. (If you don't reach 0 or 1, of course, the division or log doesn't exist in the natural numbers.) Finding the roots of natural numbers takes guesswork or search.
In the real numbers it's reversed; there's convenient algorithms for finding roots, which you can find and verify without needing calculus (though calculus gives a proof), while finding logs needs tables of pre-calculated values.
UPDATE: Well, findings logs is a generalization of the natnum procedure; instead of dividing by your base, you divide by powers of your base. E.g in finding log_10 (2), you divide 2 by the fourth root of 10 (the square root is too big), then by the 32nd root of 10, and so on, and the log is the sum of the root powers (1/4 + 1/32 +...) You need the table of the various roots of 10.
Or, of course, you could just do trial and error. "Third root of 10 is bigger than 2, fourth root is smaller, so the log must be a rational in between..." It'll help to use the root algorithms, rather than needing trial and error on those, but it'd still be a fair bit of work.