Foundations — Change of base formula — proof

Before you can prove the change of base formula in Change of base formula — proof, every symbol in it must already feel obvious. This page builds each one from absolute zero, in the order they lean on each other.

0. What a "power" even is

Everything below rests on one tiny idea, so we start there.

The picture: think of a staircase where each step multiplies your height by $a$. After $n$ steps you're at height $a^n$.

1. The symbols $a$, $b$, $x$ — bases and arguments

The formula uses three letters. They are not interchangeable; each plays a fixed role.

Why $>0$ and $\ne 1$?

• Base $\le 0$: you can't consistently raise a negative or zero base to every power (what is $(-2{)}^{0.5}$? not a real number). So bases must be positive.
• Base $=1$: $1^n=1$ for every $n$. So "how many 1s make 8?" has no answer — $1$ can never reach $8$. A base of $1$ breaks the whole idea.
• Argument $x>0$: a positive base raised to any real power is always positive, so it can only ever reach positive numbers. You can never get $x=-5$ or $x=0$.

2. The logarithm ${\log }_{a}x$ — a question, not an operation

This is the heart of everything.

So ${\log }_{2}8$ asks "what power of $2$ gives $8$?" — and since $2^3=8$, the answer is ${\log }_{2}8=3$.

The picture: a logarithm and an exponential are the same arrow read in opposite directions.

Let's read the symbol carefully so notation never trips you:

$\underset{\text{"which power"}}{\underbrace{\log }}\underset{\text{base (small, low)}}{\underbrace{{}_2}}\underset{\text{argument (the target)}}{\underbrace{8}}$

3. Two special log values you must reflexively know

The proof and its examples quietly use two facts. Build them now so they're free later.

Read them as questions:

• ${\log }_{a}a$: "what power of $a$ gives $a$?" → clearly $1$.
• ${\log }_{a}1$: "what power of $a$ gives $1$?" → anything to the $0$ is $1$, so $0$.

4. The power law — the engine of the proof

The proof's crucial move (Step 4) is a law of logs. Meet it here.

Why is this true? Because a log measures "how many multiplications". If $a$ needs ${\log }_{b}a$ base-$b$ steps to build, then $a^y$ (that block used $y$ times) needs $y$ times as many steps.

5. ${\log }_{10}$, $\ln$, and $e$ — the "calculator rulers"

The formula exists because of a practical limit: real calculators only carry two log buttons.

6. The fraction bar and "$\ne 0$" — why division is safe

Chaining it all together: we divide by ${\log }_{b}a$ in the proof. From Section 3, ${\log }_{b}a=0$ only if $a=1$. Section 1 already banned $a=1$. Therefore ${\log }_{b}a\ne 0$, and the division is always legal. Every restriction earns its keep.

Prerequisite map

Equipment checklist

Cover the right side and test yourself — if any answer is shaky, reread that section before the proof.

Connections

• Change of base formula — proof — the topic every symbol here feeds into.
• Definition of a logarithm — the single fact the whole proof rests on.
• Laws of logarithms — home of the power law engine.
• Exponential functions — where $a^y$ and $e^x$ live.
• Natural logarithm ln — the base-$e$ calculator ruler.
• Solving exponential equations — where dragging exponents down pays off.