Graphs of logarithmic functions

WHAT is a logarithmic function?

WHY $a\neq 1$ and $a>0$ ? Because $1^y=1$ for every $y$ (no unique inverse), and negative bases give undefined values for many exponents. Same restriction as exponentials — logs inherit it.

HOW the graph is built (derivation from scratch)

We already know the exponential graph $y=a^x$ (for $a>1$ ):

passes through $(0,1)$
has a horizontal asymptote $y=0$
always positive, increasing.

To get $y=\log_a x$ we reflect in $y=x$ (because that is what "inverse function" means geometrically — swap input and output). Reflecting a point $(p,q)$ in $y=x$ gives $(q,p)$ . So every feature swaps its roles:

Feature of $y=a^x$	Swap $x\leftrightarrow y$	Feature of $y=\log_a x$
through $(0,1)$		through == $(1,0)$ ==
horizontal asymptote $y=0$		vertical asymptote == $x=0$ ==
defined for all $x$		defined only for $x>0$
range all $y>0$		range all real $y$
through $(1,a)$		through $(a,1)$

Figure — Graphs of logarithmic functions

Shape depending on the base

WHY does it rise so slowly? To increase $y$ by $1$ , you must multiply $x$ by $a$ . To get $y=6$ with base $10$ you need $x=10^6=1000000$ . Big input → small output. This is the whole reason logs are used to "compress" huge ranges (decibels, pH, Richter scale).

Transformations of $\log$ graphs

Using log laws we can rewrite transformed graphs as vertical shifts:

$\log_a(kx) = \log_a k + \log_a x$

So multiplying inside by a constant $k$ is the same as shifting the whole curve up by $\log_a k$ . That's a neat, testable fact.

Standard transformations of $y=\log_a x$ :

$y=\log_a x + c$ → shift up $c$ .
$y=\log_a(x-b)$ → shift right $b$ , asymptote moves to $x=b$ .
$y=-\log_a x$ → reflect in $x$ -axis.

Worked examples

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

A logarithm is a "how many times do I multiply" machine. $\log_{10}1000$ asks "how many 10s multiplied make 1000?" → three. Now draw a picture of it: the graph starts way down low near the left edge (you can never touch the up-and-down line at $x=0$ ), crosses the bottom at $x=1$ (because $\log$ of 1 is always 0 — you multiply zero times to get 1), then climbs up but gets lazier and lazier, needing 10× more input for each extra step. It's literally the exponential graph looked at in a mirror placed along the diagonal line.

Active recall

What transformation maps

y=a^x

onto

y=\log_a x

Reflection in the line

y=x

(inverse function).

Through which fixed point does every

y=\log_a x

pass?

(1,0)

, since

\log_a 1=0

What is the asymptote of

y=\log_a x

and its type?

x=0

, a vertical asymptote.

State the domain and range of

y=\log_a x

(

a>1

Domain

x>0

; range all real numbers.

Why is there no

y

-intercept for

y=\log_a x

Because

x=0

is excluded (asymptote); log of 0 is undefined.

For

a>1

, is

\log_a x

increasing or decreasing, and how fast?

Increasing, but slowly —

x

must be multiplied by

a

to raise

y

by 1.

What point (other than

(1,0)

) always lies on

y=\log_a x

(a,1)

, since

\log_a a=1

Describe the graph of

y=\log_a(x-b)

y=\log_a x

shifted right by

b

; asymptote at

x=b

Rewrite

\log_a(ax)

as a shift.

\log_a x + 1

(shift up 1), using

\log_a(ax)=\log_a a+\log_a x

Solve

\log_{10}x=3

from the graph reasoning.

x=10^3=1000

(rewrite as

x=a^y

Connections

Exponential functions and their graphs — the reflection partner.
Inverse functions and reflection in y=x — the geometric reason for the shape.
Laws of logarithms — turn inside-changes into vertical shifts.
Natural logarithm ln and e — the special base $a=e$ .
Solving equations with logarithms — reading intersections off the graph.
Transformations of graphs — shifts, reflections, stretches applied here.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, logarithm basically exponential ka ulta (inverse) hai. Agar $y=a^x$ ka graph tumhe pata hai, to $y=\log_a x$ ka graph banane ke liye bas usko $y=x$ line mein mirror kar do. Jaha exponential $(0,1)$ se guzarta hai, waha log $(1,0)$ se guzrega — bas $x$ aur $y$ swap ho jaate hain. Yehi ek trick se poora graph samajh aa jaata hai.

Do cheezein hamesha yaad rakho: log sirf $x>0$ ke liye define hota hai (negative ya zero ka log nahi hota), aur $x=0$ pe ek vertical asymptote hoti hai — curve niche $-\infty$ ki taraf bhaagta hai par $y$ -axis ko chhoota nahi. Exponential mein horizontal asymptote thi, mirror karne pe wo vertical ban gayi. Isliye log graph ka koi $y$ -intercept nahi hota, sirf $x$ -intercept $(1,0)$ hota hai.

Log dheere-dheere badhta hai kyunki $y$ ko $1$ badhane ke liye $x$ ko $a$ se multiply karna padta hai. Isliye $\log_{10}1000000=6$ — bada input, chhota output. Yahi property real life mein kaam aati hai (pH, decibel, Richter scale) jaha bahut badi range ko compress karna hota hai.

Transformations mein sirf ek jaal hai: andar ka change horizontal hota hai, bahar ka vertical. $\log(x-3)$ matlab right shift 3 (asymptote $x=3$ pe), jabki $\log x + 3$ matlab up shift 3. Aur ek smart baat — $\log_a(ax)=\log_a x+1$ , yaani base ka multiply andar, wo actually curve ko upar shift kar deta hai. Exam mein yeh point aksar poochha jaata hai.

WHAT is a logarithmic function?

HOW the graph is built (derivation from scratch)

Shape depending on the base

Transformations of $\log$ graphs

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Exponentials & Logarithms

Connections