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M01.4 · Exponentials and logarithms

A free, illustrated module of 4 lessons and one interactive in M01 Math Foundations: Powers and their rules; Exponential growth and decay; Logarithms: the exponent as the answer; Log scales and why machine learning takes logs.

M01.4 L0 · Foundation 4 lessons · about 100 min Live · in review

Part of M01 Math Foundations in the Mathematics track.

Multiplying by the same number again and again is behind compound growth, decaying learning rates, memory sizes and the tiny probabilities of machine learning. These four lessons count factors to find the rules of powers, turn a power into the exponential function with its growth, decay and the number e, run it backwards as the logarithm, and use logarithms to draw rulers that show a millionth and a million together and to turn long products into sums that a computer can hold.

What you need first. Arithmetic with fractions and decimals, whole-number powers and square roots (M01.1), and functions, graphs and inverse functions (M01.3). The code uses loops, functions, lists and importing a module, from P01.2 to P01.6.

Lessons

1. Powers and their rules M01.4.1 25 min

   A power is a count of equal factors. Derive the product, quotient and power-of-a-power rules by counting, see why zero, negative and fractional exponents must mean what they do, and read scientific notation such as Python’s 3e-4.

2. Exponential growth and decay M01.4.2 25 min

   A line adds the same amount at every step; an exponential multiplies by the same factor. Meet the exponential function, its growth and decay, doubling times and half-lives read from a table, and the number e as the value that compounding in ever more steps settles towards.

3. Logarithms: the exponent as the answer M01.4.3 25 min

   A logarithm answers one question: to what power must the base be raised to give this number? Meet the logarithm as the inverse of the exponential, derive its product, quotient and power laws and the change-of-base formula, and find the doubling time of 5 percent growth exactly.

4. Log scales and why machine learning takes logs M01.4.4 25 min

   On a log scale every equal step multiplies by the same factor, so one ruler can show 0.0001 and 1,000,000 together. Learn to read log axes and orders of magnitude, and see why machine learning adds logarithms instead of multiplying probabilities.

The interactive

M01.4-X1 · Log ruler. Bend a ruler from linear to logarithmic and watch the crowded values from 1 to 1,000,000 spread out into equal steps. Then choose a and b and lay their lengths end to end: on the log ruler they land on a times b, as on a slide rule. It lives in Log scales and why machine learning takes logs.