Log 1 Is Zero — and Why That's the Whole Foundation

The One Identity to Rule Them All

If you remember nothing else from this track, remember this:

$lo g_{a} (1) = 0 for every base a$

Why? Because $a^{0} = 1$ for every base $a$ (any non-zero number raised to the zero power equals 1). And the logarithm is just asking: "to what power do I raise the base to get this?" If the answer is "the zero power" (no multiplication at all), the result is 1.

This little fact is the anchor of every log identity you'll ever use. Once you internalize it, the rest of logarithms becomes navigation rather than memorization.

The Counterclockwise Trick

Reading $lo g_{b} (x) = y$ — the textbook way says "log base b of x equals y." Dad's trick: read it counterclockwise. Start at the base $b$ , move to the result $y$ , arrive at the argument $x$ . The relationship: " $b$ raised to the $y$ gives $x$ ", i.e. $b^{y} = x$ . Logs and exponentials are mirror images.

Two Bases You'll See in AI

Base 10 (common log) — written as $lo g (x)$ . Used in human-facing scales: decibels, pH, earthquake magnitudes.
Base $e$ (natural log) — written as $ln (x)$ . $e \approx 2.71828$ is the math constant of continuous growth. Calculus loves base $e$ because the derivative of $e^{x}$ is $e^{x}$ itself. AI loves it because most loss functions and activation functions involve it.

Log of 1 is 0. Anchor for everything else. Logs and exponentials are inverse operations — one undoes the other. Base

e

is calculus's home base; base 10 is humanity's.

Code

The identity in three bases·python

import numpy as np

# Log of 1 is 0 — every base
print(np.log(1))      # 0.0  (natural log, base e)
print(np.log10(1))    # 0.0  (base 10)
print(np.log2(1))     # 0.0  (base 2)

# Inverses: e^(ln(x)) = x for x > 0
print(np.exp(np.log(5.0)))   # 5.0

# Common logs in AI
print(np.log(np.e))          # 1.0 — defining property of natural log

Log 1 Is Zero — and Why That's the Whole Foundation

The One Identity to Rule Them All

The Counterclockwise Trick

Two Bases You'll See in AI

Code

Exercise

Progress

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