Graphs of logarithmic functions: Foundations — Maths deep dive

Before you can read the parent note Graphs of logarithmic functions without tripping, you need to own every symbol it throws at you — including the base $a$ and the notation $lo g_{a} x$ , which we build up carefully below. This page names each symbol, draws the picture behind it, and says why the topic can't proceed without it. Read top to bottom — each block uses only what came before.

0. The most basic bricks: $x$ , $y$ , and a point $(x, y)$

Figure — Graphs of logarithmic functions

Reveal-check:

The point

(3, - 2)

sits where?

3 right of centre, 2 down.

1. A function $y = f (x)$ : the input → output machine

The parent topic is all about two machines: the exponential machine $a^{x}$ and the logarithm machine $lo g_{a} x$ (both are defined below, once the base $a$ is built). Everything else is comparing them.

Reveal-check:

Why can't a function give two outputs for one input?

Then it wouldn't be a function — one input, one answer.

2. The power notation $a^{x}$ : repeated multiplying, then everywhere in between

Intuition Stretching the exponent to

any real number "Multiply $x$ times" only makes sense when $x$ is a whole number — you can't multiply "half a time". But we still want $a^{1/2}$ , $a^{0.7}$ , even $a^{2}$ , so the curve has no gaps. The rules below force the meaning of these in-between exponents, and the picture is simply "fill in the dots smoothly between the whole-number points":

Roots fill the fractions: $a^{1/2} = a$ , because $(a^{1/2})^{2} = a^{1} = a$ — the power that squares to $a$ is the square root. Likewise $a^{1/3} = 3 a$ .
Any decimal sits between two fractions, and we take the value the fractions close in on. So $a^{0.7}$ is squeezed between $a^{7/10}$ and nearby roots.

The upshot: $a^{x}$ has a sensible value for every real $x$ , and joining them gives one unbroken curve. This "smooth fill-in" is what lets us draw $y = a^{x}$ as a solid line in §4.

From $a^{0} = 1$ and $a^{1} = a$ we get the exponential-graph anchors $(0, 1)$ and $(1, a)$ , and from $a^{- 1} = 1/ a$ the anchor $(- 1, \frac{1}{a})$ . (These are points on $y = a^{x}$ . Their reflected twins on the log-graph — $(1, 0)$ , $(a, 1)$ , $(\frac{1}{a}, - 1)$ — arrive once we meet the logarithm in §6–7; don't worry about them yet.)

Reveal-check:

What is

1 0^{0}

1

What is

5^{- 1}

\frac{1}{5}

What is

9^{1/2}

and why?

3

, because

3^{2} = 9

— the half-power is the square root.

3. Why $a > 0$ and $a \neq = 1$

Reveal-check:

Why is base

a = 1

banned?

1^{x} = 1

always, so it can't be reversed to a unique

x

4. The exponential graph $y = a^{x}$ — both base cases

Its shared signature features (all provable from §2):

passes through $(0, 1)$ — because $a^{0} = 1$ ;
always positive — recall from §2 that $a^{x} > 0$ for every real $x$ , so the curve never touches or crosses the $x$ -axis;
a horizontal asymptote at $y = 0$ (defined next) — approached on the left if $a > 1$ , on the right if $0 < a < 1$ .

This two-case split feeds straight into the parent note's "increasing vs decreasing log" section: reflecting a rising exponential gives a rising log; reflecting a falling one gives a falling (decreasing) log. See Exponential functions and their graphs.

Reveal-check:

Where does

y = a^{x}

cross the

y

-axis, for any base?

(0, 1)

y = (\frac{1}{2})^{x}

rising or falling?

Falling (decaying), since

0 < a < 1

5. Asymptote: the line the curve chases but never touches

Reveal-check:

What does "asymptote" mean in one phrase?

A line the curve approaches but never touches.

6. The logarithm $lo g_{a} x$ : an exponent asked backwards

Reveal-check:

What question does

lo g_{a} x

answer?

"What power of

a

gives

x

Rewrite

lo g_{a} x = y

without a log.

x = a^{y}

7. Inverse & reflection in $y = x$ : the picture behind the graph

For the deeper "why", see Inverse functions and reflection in y=x.

Reveal-check:

Reflecting

(0, 1)

y = x

gives what point?

(1, 0)

Which line is the mirror between

a^{x}

and

lo g_{a} x

y = x

For

0 < a < 1

, is

lo g_{a} x

rising or falling?

Falling (decreasing).

8. Domain and range: the allowed inputs and outputs

Reveal-check:

Domain of

lo g_{a} x

x > 0

Range of

lo g_{a} x

All real numbers.

9. The special bases $10$ and $e$

You'll meet $e$ properly in Natural logarithm ln and e. For now: $ln x$ is just $lo g_{a} x$ with $a = e$ , nothing new.

Reveal-check:

What base does

ln x

use?

e \approx 2.718

10. Two tools the parent note reuses

Reveal-check:

lo g_{a} (k x)

equals what sum, and what must

k

satisfy?

lo g_{a} k + lo g_{a} x

, with

k > 0

lo g_{a} k

exists.

f (x - b)

moves the graph which way?

Right by

b

f (x + b)

moves the graph which way?

Left by

b

Prerequisite map

Equipment checklist

Tick each only when you can answer without peeking.

Place the point

(- 2, 3)

in words.

2 left of centre, 3 up.

State what a function guarantees about outputs.

Exactly one output per input.

Evaluate

3^{0}

3^{1}

3^{- 1}

1

3

\frac{1}{3}

What does

a^{1/2}

mean and why is it positive?

a

; the root of a positive number is positive.

Why is

a^{x} > 0

for every real

x

It is built from products, reciprocals and roots of the positive base

a

Why must the base satisfy

a > 0, a \neq = 1

Base 1 is stuck on 1; negative bases give undefined roots — no reversible machine.

Name the features of

y = a^{x}

and how the two base cases differ.

Through

(0, 1)

, always positive, horizontal asymptote

y = 0

; rises if

a > 1

, falls if

0 < a < 1

Define "asymptote".

A line the curve approaches but never touches.

Turn

lo g_{a} x = y

into power form.

x = a^{y}

Reflecting

(p, q)

y = x

gives what?

(q, p)

For

0 < a < 1

, does the log rise or fall?

Falls — it's the reflection of a decaying exponential.

Give the domain and range of

lo g_{a} x

Domain

x > 0

; range all reals.

What are the bases of

l g x

and

ln x

10

and

e

State

lo g_{a} (k x)

as a sum and the condition on

k

lo g_{a} k + lo g_{a} x

, requires

k > 0

Which way does

f (x - b)

shift, and

f (x) + c

f (x - b)

right by

b

;

f (x) + c

up by

c

Connections

Exponential functions and their graphs — the reflection partner built in §4.
Inverse functions and reflection in y=x — the geometry of §7.
Laws of logarithms — the product rule of §10.
Natural logarithm ln and e — the base $e$ of §9.
Solving equations with logarithms — using the switch trick of §6.
Transformations of graphs — inside vs outside from §10.