Visual walkthrough — Graphs of logarithmic functions

Before we draw anything, we agree on three words:

We build with the friendly base $a=2$, then generalise. Everything below uses only "multiply", "power", and "swap".

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Step 1 — Plot the exponential we already trust

WHAT. We start with $y=2^x$ and mark four honest points.

$\underset{\text{height}}{\underbrace{y}}=\underset{\text{base }a}{\underbrace{2}}{}^{\stackrel{\text{power}}{\overbrace{x}}}$

Put in $x=-1,0,1,2$:

$2^{-1}=\frac{1}{2},2^0=1,2^1=2,2^2=4$

So the dots are $(-1,\frac{1}{2}),(0,1),(1,2),(2,4)$.

WHY. We cannot draw the log directly — its whole meaning is "the reverse of this exponential". So we must have the exponential physically on the paper first, as the thing we will flip.

PICTURE. The blue curve hugs a horizontal wall at $y=0$ on the left (it never touches it) and shoots up on the right. Notice the anchor dot at $(0,1)$ — this is the one that will matter most.

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Step 2 — Draw the mirror line $y=x$

WHAT. Add the diagonal line where "across" equals "up": the set of points $(t,t)$.

$y=x⟺\text{height}=\text{sideways distance}$

WHY. Reflecting a function in this exact line is the geometric meaning of "inverse". Here is the reason in one breath: a function sends input $\to$ output horizontally then vertically; its inverse must send output $\to$ input, i.e. do the vertical and horizontal moves in the opposite order. Swapping the order of those two moves is exactly what a flip across $y=x$ does. (More on this in Inverse functions and reflection in y=x.)

PICTURE. The pale-yellow diagonal cuts the plane in half. Any point and its mirror image sit at equal perpendicular distance on opposite sides — like folding the paper along the yellow line.

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Step 3 — Reflect the four anchor points (swap the numbers)

WHAT. Take each exponential dot $(p,q)$ and swap to $(q,p)$:

on $y=2^x$	swap $x\leftrightarrow y$	on $y={\log }_{2}x$
$(-1,\frac{1}{2})$		$(\frac{1}{2},-1)$
$(0,1)$		$(1,0)$
$(1,2)$		$(2,1)$
$(2,4)$		$(4,2)$

WHY. If $2^p=q$ then by the very definition of log, ${\log }_{2}q=p$. So the point $(q,p)$ must lie on the log graph. Swapping coordinates isn't a trick — it is the definition acting on points.

$2^0=1⟹{\log }_{2}1=0\text{(the dot }(0,1)\text{ becomes }(1,0)\text{)}$

PICTURE. Each blue dot fires an arrow straight across the yellow mirror to its pink partner. Watch $(0,1)$ land on $(1,0)$ — the exponential's $y$-intercept becomes the log's $x$-intercept.

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Step 4 — Join the mirrored dots into the log curve

WHAT. Connect the pink dots smoothly to reveal $y={\log }_{2}x$.

WHY. The exponential was a smooth unbroken curve, and reflection can't tear it — so the mirrored dots must also lie on one smooth curve. Joining them is legitimate, not guessing.

PICTURE. The pink curve rises from bottom-left, crosses the floor at $(1,0)$, and climbs lazily to the right. Compare its lazy climb with the blue curve's steep climb: they are the same curve, just viewed with $x$ and $y$ swapped.

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Step 5 — What happened to the wall? (the asymptote swaps direction)

WHAT. The exponential's horizontal wall $y=0$ becomes the log's vertical wall $x=0$.

$\underset{\text{floor of }{2}^x}{\underbrace{y=0}}\stackrel{\text{swap }x\leftrightarrow y}{\to }\underset{\text{left wall of }{\log }_{2}x}{\underbrace{x=0}}$

WHY. A horizontal line $y=0$ reflected in $y=x$ becomes the vertical line $x=0$ — swapping the roles of the two axes swaps "horizontal barrier" for "vertical barrier". So as $x\to 0^{+}$ (creeping toward the wall from the right), the log dives to $-\infty$; it never touches the wall, just as $2^x$ never touched its floor.

Every case near the wall:

• $x=\frac{1}{2}=2^{-1}\Rightarrow y=-1$
• $x=\frac{1}{4}=2^{-2}\Rightarrow y=-2$
• $x=\frac{1}{1000}\approx 2^{-10}\Rightarrow y\approx -10$

Halving $x$ drops $y$ by exactly $1$ — forever downward, no bottom.

PICTURE. The dashed vertical wall at $x=0$; the pink curve slides down alongside it toward $-\infty$, mirroring the dashed horizontal floor the blue curve rode.

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Step 6 — Edge case: the left half of the plane is empty

WHAT. There is no curve for $x\le 0$. No point of $y={\log }_{2}x$ ever sits on or left of the wall.

WHY. Every mirrored dot came from an exponential dot, and $2^p$ is always positive. Positive $q$ values only. So after the swap, every $x$-coordinate is positive — the domain is $x>0$. You cannot ask "what power of $2$ gives a negative number (or zero)?", because no power of $2$ is ever $\le 0$.

$\text{no }y\text{ with }{2}^y=0⟹{\log }_{2}0\text{ undefined};2^y=-3\text{ impossible}⟹{\log }_2(-3)\text{ undefined}$

PICTURE. The whole left region is shaded out ("forbidden zone"); the exponential's missing region (below its floor $y<0$... wait — below floor) maps to this. The pink curve lives strictly to the right of the wall.

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Step 7 — Case split by base: $a>1$ versus $0<a<1$

WHAT. With $a>1$ the log rises. With $0<a<1$ it falls. The switch:

${\log }_{1/a}x=\frac{{\log }_{a}x}{{\log }_a(1/a)}=\frac{{\log }_{a}x}{-1}=-{\log }_{a}x$

WHY. ${\log }_a(1/a)={\log }_a{a}^{-1}=-1$ (a law of logs — see Laws of logarithms). Dividing by $-1$ flips the curve upside down. So a shrinking base is just the growing-base log reflected in the $x$-axis. Same wall $x=0$, same crossing $(1,0)$ — only the climbing direction reverses.

• Base $a>1$ (e.g. $2$, $10$, $e$): through $(1,0)$ and $(a,1)$, increasing.
• Base $0<a<1$ (e.g. $\frac{1}{2}$): through $(1,0)$ and $(a,1)$, decreasing.

PICTURE. Two pink curves crossing at $(1,0)$: one climbs (base $2$), its mirror-in-the-floor twin descends (base $\frac{1}{2}$). The dashed horizontal $x$-axis is the mirror between them.

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Step 8 — Degenerate bases: why $a=1$ and $a\le 0$ are banned

WHAT. No log graph exists for $a=1$, or $a\le 0$.

WHY.

• $a=1$: $1^y=1$ for every $y$. So "${\log }_{1}3=?$" has no answer ($1^y$ never reaches $3$), and "${\log }_{1}1=?$" has infinitely many answers. No single-valued curve possible — the exponential $y=1^x$ is a flat line $y=1$, and a flat line has no reflection that is a function.
• $a\le 0$: $a^x$ isn't even defined for many $x$ (e.g. $(-2{)}^{1/2}$), so there is no clean exponential to reflect.

PICTURE. The flat line $y=1^x$ (constant) reflected in $y=x$ becomes the vertical line $x=1$ — which fails the "one output per input" test, shown by a red cross.

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The one-picture summary

Everything at once: the blue exponential $y=2^x$, the yellow mirror $y=x$, the pink log $y={\log }_{2}x$, both walls, and the shared anchors.

Recall Feynman retelling of the whole walkthrough

I drew the exponential $2^x$ first — the curve that doubles every step and rides a floor at $y=0$. Then I laid down the diagonal mirror $y=x$. Reflection just swaps a point's two numbers, and by the definition of a log that swap sends every exponential dot straight onto the log curve — so $(0,1)$ became $(1,0)$. Joining the mirrored dots gave the pink log curve: crosses the floor at $x=1$, climbs lazily. The exponential's flat floor became the log's tall left wall at $x=0$ — that's why logs plunge to $-\infty$ near zero and never reach the wall. Because $2^x$ is always positive, no dot ever lands left of the wall, so logs only eat positive numbers. Bigger-than-one bases climb; between-zero-and-one bases are the same curve flipped upside-down; base $1$ or negative bases give no honest curve at all. One diagonal mirror explains the entire shape.

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Connections

• Exponential functions and their graphs — the curve we reflected in Step 1.
• Inverse functions and reflection in y=x — why the mirror is $y=x$ (Step 2).
• Laws of logarithms — the flip ${\log }_{1/a}x=-{\log }_{a}x$ (Step 7).
• Natural logarithm ln and e — same walkthrough with base $a=e$.
• Transformations of graphs — shifting and reflecting the finished curve.
• Solving equations with logarithms — reading crossings off this picture.

Concept Map