Change of base formula — proof — Maths notes

What the formula says

WHAT each piece is:

$a$ = the old base (the one you want to get rid of).
$b$ = the new base (the one your calculator understands, e.g. $10$ or $e$ ).
$x$ = the argument (the number inside).

Derivation from scratch (never memorise — build it)

HOW we derive it, step by step:

Step 1 — Name the thing we want. Let $y = \log_a x.$ Why this step? We can't manipulate a mystery; giving it a name $y$ lets us do algebra on it.

Step 2 — Rewrite in exponential form. By the definition above, $a^y = x.$ Why this step? Exponentials are easier to attack with logs of any base, because logs turn powers into products.

Step 3 — Take $\log_b$ of BOTH sides. $\log_b(a^y) = \log_b x.$ Why this step? Applying the same function to both sides keeps equality. We choose base $b$ because that's the base we're allowed to use (calculator base).

Step 4 — Use the power law $\log_b(a^y)=y\log_b a$ . $y\,\log_b a = \log_b x.$ Why this step? This is the whole point of taking a log: it drags the exponent $y$ down into a coefficient, so we can isolate it.

Step 5 — Solve for $y$ (divide by $\log_b a$ , which is $\neq 0$ since $a\neq 1$ ): $y = \frac{\log_b x}{\log_b a}.$

Step 6 — Recall what $y$ was. Since $y=\log_a x$ , $\boxed{\log_a x = \frac{\log_b x}{\log_b a}}\qquad\blacksquare$

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

A log is a question: "How many times do I multiply this base to reach the number?" $\log_2 8$ asks "how many 2s multiplied make 8?" — answer 3. Now imagine you only own a base-10 ruler to measure powers. You can still measure the "2-question" — just measure how tall $8$ is in base 10, measure how tall $2$ is in base 10, and divide. The scaling cancels out and you get the true answer, 3. Same height, different ruler.

Active recall

What does the definition

\log_a x = y

mean in exponential form?

a^y = x

State the change of base formula.

\log_a x = \dfrac{\log_b x}{\log_b a}

In change of base, which quantity goes in the numerator?

The log of the argument

x

(in the new base).

Which quantity goes in the denominator?

The log of the old base

a

(in the new base).

Why is dividing by

\log_b a

allowed?

Because

a \neq 1

, so

\log_b a \neq 0

First step of the proof?

Let

y=\log_a x

, then rewrite as

a^y=x

Which log law makes the exponent come down in the proof?

The power law

\log_b(a^y)=y\log_b a

Prove the reciprocal identity

\log_a b = 1/\log_b a

Set

x=b

\log_a b=\frac{\log_b b}{\log_b a}=\frac{1}{\log_b a}

Compute

\log_2 8

via base 10.

\frac{\log 8}{\log 2}=3

Does the new base have to be 10 or

e

No — any valid base

b>0,\ b\neq1

works; 10/

e

are just convenient.

Simplify

\log_a b\cdot\log_b a

=1

(they are reciprocals).

Connections

Definition of a logarithm — the single fact the whole proof rests on.
Laws of logarithms — the power law is the engine of the derivation.
Exponential functions — Step 2 converts log→exponential form.
Natural logarithm ln — the usual "new base" $b=e$ .
Solving exponential equations — change of base lets you isolate unknown exponents.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, logarithm basically ek sawaal hai: "base ko kitni power do taaki number mil jaaye?" Jaise $\log_2 8$ ka matlab "2 ko kitni baar multiply karun ki 8 aaye?" — answer 3. Ab problem yeh hai ki calculator me sirf $\log_{10}$ aur $\ln$ ke buttons hote hain, base 2 ka button nahi hota. Toh change of base formula humara translator hai jo kisi bhi base ko 10 ya $e$ me convert kar deta hai.

Formula hai $\log_a x = \frac{\log_b x}{\log_b a}$ . Yaad rakhne ka simple tareeka: argument ( $x$ ) upar, purana base ( $a$ ) neeche, naya base ( $b$ ) dono me. Iska proof rattne ki zarurat nahi — sirf ek fact se banta hai: $\log_a x = y$ ka matlab $a^y = x$ . Bas $y$ ko naam do, exponential form me likho, dono taraf $\log_b$ lagao, power law se $y$ ko neeche laao, aur divide karke $y$ nikaal lo. Ho gaya!

Sabse bada intuition (dual coding wala diagram dekho): chahe ruler base 2 ka ho ya base 10 ka, height same rehti hai — sirf measure karne ka scale badalta hai, aur woh scaling division me cancel ho jaati hai. Isliye answer nahi badalta.

Common galti se bacho: log ko $\frac{\log a}{\log x}$ me todna galat hai (single $\log_a$ ka koi matlab nahi hota), aur argument hamesha upar jaata hai. Ek quick check: $\log_2 8 = 3$ hona chahiye, toh $\frac{\log 8}{\log 2}$ (bada upar) hi sahi hai, ulta nahi.

Change of base formula — proof

What the formula says

Derivation from scratch (never memorise — build it)

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Exponentials & Logarithms

Connections