Maths 1.2Lesson outlineLesson 2 of 3 in Module 1

Exponents and Logarithms

Know why logs appear everywhere in ML: they turn products into sums, and huge ranges into small ones.

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The words this lesson introduces, each in one line. The module’s glossary collects them all.

base**, **exponent: in bⁿ, b is the base and n counts how many times it is multiplied | 2³ = 2 × 2 × 2
power: bⁿ | 2⁵ = 32
root: b^(1/n), the number whose nth power is b | 2^½ = 1.4142
e: the base whose exponential has slope one at zero, 2.71828… | slope 1.0005 at step 0.001
exponential function: exp(x) = eˣ | np.exp(1) = 2.718281828459045
logarithm: the exponent that gives a number | log₁₀ 1000 = 3
natural log ln: the logarithm with base e | ln 0.5 = −0.6931
inverse function: a function that undoes another; its graph is the reflection across y = x | ln undoes exp
product rule: log(ab) = log a + log b | ln 0.1 = −2.3026
scientific notation: a number times a power of ten; Python writes it 3.6e-06 | 3.6 × 10⁻⁶
log scale: an axis where equal distances are equal ratios | 1, 0.1, 0.01 evenly spaced
odds: p / (1 − p) | 4 for p = 0.8
log-odds (logit): ln(p / (1 − p)); the sigmoid turns it back into a probability | 1.3863 for p = 0.8
common mistake: log(a + b): the rule is for a product; log(a + b) is not log a + log b | ln 0.7 = −0.3567, not −2.3026
common mistake: np.log of zeros: gives -inf with a warning, not an error; use np.log1p | 108 cells of the email counts