Foundations — Logarithm — definition as inverse of exponential

Before you can read a single line of the parent note Logarithm — definition as inverse of exponential, you must own every symbol it quietly assumes. This page builds them in order, each from the one before, each with a picture. Never do we use a mark on the page before it means something.

1. What a base and an exponent actually are

Look at the picture below. We are not adding — we are stretching. Each press of the button multiplies the running total by the base $b=2$.

• Read left to right: $1\to 2\to 4\to 8$. The exponent is the label under each dot: $0,1,2,3$.
• $b^0=1$ because pressing the button zero times leaves you where you started: at $1$. This is not a special rule to memorise — it falls straight out of the picture.

2. The equals sign, variables, and "$⟺$"

The parent note's central line ${\log }_{b}y=x⟺b^x=y$ uses $⟺$ to say: the log statement and the exponential statement are twins. Neither is "more true"; they are one fact seen from two sides. You need $⟺$ before that line makes sense, so we install it now.

3. Positive, negative, and fractional exponents

The next figure continues the doubling machine in both directions, so every case is covered — not just the tidy whole numbers.

• Going right (exponent up): $8,16,32$ — the values shoot upward.
• Going left past $0$ (exponent negative): $\frac{1}{2},\frac{1}{4},\frac{1}{8}$ — the values shrink toward $0$ but never touch or cross it.
• Notice the height is always positive. This single visual fact is why the parent restricts ${\log }_b$ to positive inputs: you can never land on $0$ or below by doubling from $1$.

4. Why $b>0$ and $b\ne 1$ — seen, not stated

• Left panel ($b=1$): every press multiplies by $1$, so the value never moves — a flat line at height $1$. Ask "$1$ to what power gives $5$?" and the answer is nowhere. A flat line can't be run backwards to find a unique exponent, so no log base $1$ can exist.
• Right panel ($b=-2$): the value jumps between positive and negative ($1,-2,4,-8,\dots$). It's a scatter of dots, not a smooth climbing curve. There is no single clean exponent for an in-between value — so negative bases are banned too.
• A healthy base ($b>0,b\ne 1$) gives a smooth curve that climbs (or falls) without ever repeating a height. Only such a curve can be reversed — and reversing it is the logarithm.

5. Function, inverse, and "one-to-one"

Only a one-to-one machine can be undone — otherwise the undo button wouldn't know which input to return to. From §4 we saw $b^x$ (with a healthy base) climbs without repeating: it is one-to-one, so its undo button exists. We name that undo button ${\log }_b$.

6. Reflection in the line $y=x$

• The magenta curve is $y=b^x$; it hugs the floor on the left and passes through $(0,1)$.
• The violet curve is its mirror image $y={\log }_{b}x$; it hugs the wall and passes through $(1,0)$.
• The dashed navy diagonal is the mirror $y=x$. Fold the page along it and the two curves land on each other — that folding is the inverse relationship made visible.
• Every landmark swaps: horizontal asymptote $y=0$ becomes vertical asymptote $x=0$; the point $(0,1)$ becomes $(1,0)$.

Prerequisite map

For where these tools go next, see [[Exponential functions — $b^x$ and its graph]] and Inverse functions and reflection in $y=x$; once the definition is solid you can pick up Laws of logarithms — product, quotient, power rules and Natural logarithm $\ln$ and Euler's number $e$.

Equipment checklist

Cover the right-hand side and test yourself. If any answer surprises you, reread that section before opening the parent note.