Logarithms & Properties

A logarithm is the inverse of exponentiation. $lo g_{b} (x) = y$ means $b^{y} = x$ . Logs answer the question: “what power do I raise the base to in order to get this number?”

Because $b^{y} > 0$ for every real $y$ (any positive base raised to any power stays positive), logarithms are defined only for $x > 0$ . $lo g_{b} (0)$ and $lo g_{b} (- 4)$ have no real value — they are outside the domain.

Two special bases

Common log $lo g (x)$ is base 10 (often written $lo g_{10} (x)$ ).
Natural log $ln (x)$ is base $e$ (i.e., $lo g_{e} (x)$ ).

Most ML uses natural log, because $\frac{d}{d x} [ln (x)] = \frac{1}{x}$ — the cleanest possible derivative.

Inverse properties

lo g_{b} (b^{x}) = x (log undoes the exponential)

b^{l o g_{b} (x)} = x (the exponential undoes the log)

lo g_{b} (1) = 0 (b^{0} = 1)

lo g_{b} (b) = 1 (b^{1} = b)

Internalize these. They show up everywhere.

The three core rules

Rule	Identity
Product	$lo g_{b} (x y) = lo g_{b} (x) + lo g_{b} (y)$
Quotient	$lo g_{b} (x / y) = lo g_{b} (x) - lo g_{b} (y)$
Power	$lo g_{b} (x^{k}) = k \cdot lo g_{b} (x)$

Logs convert multiplication into addition, division into subtraction, and exponentiation into multiplication. This is why logs are useful for any problem where multiplying many numbers would cause precision issues.

Solving exponential equations

Take the log of both sides (any consistent base; usually $ln$ ):

5 e^{0.2 t} + 10 5 e^{0.2 t} e^{0.2 t} 0.2 t t = 85 = 75 = 15 = ln (15) = 5 ln (15) (apply ln to both sides)

Change of base

lo g_{b} (x) = \frac{lo g _{c} ( x )}{lo g _{c} ( b )}

Useful when your calculator only does $lo g$ (base 10) and $ln$ (base $e$ ).

ML connection — log-likelihood

Most ML algorithms maximize a likelihood — the probability the model assigns to the observed data. For a dataset of $N$ independent samples, the likelihood is a product of $N$ small numbers:

L = p (x_{1}) \cdot p (x_{2}) \cdot \dots \cdot p (x_{N})

If each $p (x_{i}) < 0.01$ and $N$ is large, this product underflows to zero in floating-point arithmetic. So ML always works with the log-likelihood instead:

lo g L = lo g p (x_{1}) + lo g p (x_{2}) + \dots + lo g p (x_{N})

A sum of log-probabilities is numerically stable even for $N = 1, 000, 000$ .

ML connection — cross-entropy

The cross-entropy loss for classification:

L = - i \sum y_{i} lo g (\overset{y}{^}_{i})

Comes directly from the log-likelihood of the model being correct on the training data. The log here uses the same product/quotient rules you just drilled — when you simplify $lo g (softmax)$ for cross-entropy, the algebra cancels beautifully, giving the famously simple gradient $(\overset{y}{^} - y)$ .

You finished math-1

The 16-week journey from −(−6) to log-likelihood is over. Every concept you touched is load-bearing for the math you’ll meet in trig, calculus, linear algebra, and statistics. Take a break. The next stop on the math track (math-3, Trigonometry) builds on the function and graph intuition you developed here.

16. Logarithms & Properties

Before you start

By the end you'll be able to

Logarithms & Properties

Two special bases

Inverse properties

The three core rules

Solving exponential equations

Change of base

ML connection — log-likelihood

ML connection — cross-entropy

You finished math-1

Common mistakes

Treating $\log$ like a variable

Applying the power rule to bases instead of exponents

Forgetting the domain

Practice problems

Practice quiz

Week 16 recap

Before you start

By the end you'll be able to

Logarithms & Properties

Two special bases

Inverse properties

The three core rules

Solving exponential equations

Change of base

ML connection — log-likelihood

ML connection — cross-entropy

You finished math-1

⚠️Common mistakes

Treating $\log$ like a variable

Applying the power rule to bases instead of exponents

Forgetting the domain

Practice problems

Practice quiz

Week 16 recap

Common mistakes