Laws of logarithms — product, quotient, power rules — proofs: Foundations

Before you can prove the laws of logarithms in the parent topic, you must be able to read every mark on the page without hesitation. This page builds each symbol from absolute zero, in the order they depend on each other. Skip nothing — the proofs later re-use every single one.

1. What a "power" actually is

Look at the tower in the figure below. Each step to the right multiplies by one more copy of the base. The height of the tower is the exponent; the number written at the top is the value $b^{n}$ .

Figure — Laws of logarithms — product, quotient, power rules — proofs

Why the topic needs this: every log law is secretly a power law. If " $b^{n}$ " does not yet mean something physical to you (a growing tower), then " $lo g_{b} x = y$ " will feel like a random symbol instead of a question about that tower's height.

2. The three exponent laws (the machines the proofs run on)

Every proof in the parent note is powered by exactly three facts about powers. Meet them now, with pictures, so they are old friends when the proofs call on them.

Why "add" when we multiply? Count copies. $b^{p}$ is $p$ copies of $b$ in a row; $b^{q}$ is $q$ more copies. Stick the two rows together and you have $p + q$ copies — that is $b^{p + q}$ . The figure shows this joining of two blocks of copies.

Why "subtract" when we divide? Division cancels copies. If you have $p$ copies on top and $q$ copies on the bottom, each bottom copy cancels one top copy, leaving $p - q$ copies. Why "multiply" for a power of a power? $(b^{p})^{k}$ is " $k$ groups, each with $p$ copies" — that is $p \times k$ copies total.

Recall Why does

$b^{p} \times b^{q} = b^{p + q}$ and NOT $b^{pq}$ ? Because multiplying joins two rows of copies end-to-end, so the counts add. Multiplying the counts would mean nesting groups inside groups, which is what a power of a power does instead.

These three are proved in full in Exponentials — laws of indices; here you only need to trust and picture them, because the log proofs turn each one into a log law.

3. The logarithm — a power read backwards

The symbol " $⟺$ " means "these two statements say the exact same thing" — you may swap freely between them. That double arrow is the hinge every proof swings on: turn a log into a power, do power-algebra, turn back.

The figure shows the two-way street. Going right, the exponent form $b^{y} = x$ builds a value. Going left, the log form $lo g_{b} x = y$ asks "what exponent got me here?" Same triangle of three numbers $(b, y, x)$ , read two directions.

Why the topic needs this: the parent note's very first move in every proof is "let $p = lo g_{b} M$ , so $b^{p} = M$ ." That line is this definition. If the double arrow is not automatic, the proofs are unreadable.

4. Why the three restrictions $b > 0, b \neq = 1, x > 0$

Symbols also carry conditions, and skipping them is how students lose marks. Each restriction is a picture, not a rule to memorise.

The figure shows the curve $y = b^{x}$ hugging above the horizontal axis: it dives toward $0$ on the left but never touches it and never goes below. Every output is strictly positive — that is exactly why the argument $x$ of a log must be $> 0$ . This is the same picture you meet in Graphs of logarithmic functions.

5. Two identities that fall straight out of the definition

Because log and exponent undo each other, two "cancellation" identities appear — the parent note calls them "the key identity we lean on constantly."

Read the first aloud with the definition: " $lo g_{b} x$ is the exponent that turns $b$ into $x$ — so putting that very exponent on $b$ gives back $x$ ." The second: " $lo g_{b} (b^{k})$ asks which exponent on $b$ gives $b^{k}$ — obviously $k$ ."

6. Small notation you must not stumble on

Prerequisite map

Read it top-down: powers feed both the index laws and the definition of log; those two together produce the three log laws; the definition also forces the restrictions and gives the undo identities, which the proofs use to finish.

Equipment checklist

Cover the right side and test yourself — you are ready for the proofs only if every one is instant.

Read

b^{4}

in words

Multiply the base

b

by itself 4 times; the exponent 4 counts the copies.

State

b^{p} \times b^{q}

as a single power

b^{p + q}

— multiplying joins the rows of copies, so exponents add.

State

\frac{b ^{p}}{b ^{q}}

as a single power

b^{p - q}

— dividing cancels copies, so exponents subtract.

State

(b^{p})^{k}

as a single power

b^{p k}

— a power of a power multiplies exponents.

Rewrite

lo g_{b} x = y

in exponential form

b^{y} = x

What does

lo g_{2} 8

equal and why

3

, because

2^{3} = 8

Simplify

lo g_{b} (b^{k})

k

— the log asks "which exponent gives

b^{k}

Simplify

b^{l o g_{b} x}

x

— log and exponent are inverse (they cancel).

Why must the base satisfy

b \neq = 1

1^{y} = 1

always, so the tower never reaches other numbers — no unique exponent.

Why must the argument satisfy

x > 0

A positive base to any real power is always positive, so

0

or negatives are unreachable.

What does

⟺

mean

Both statements say the same thing; you may swap between them freely.

What does

■

mean

End of proof — we're done.

Connections

Exponentials — laws of indices — proves the three index rules this page uses as machines.
Laws of logarithms — the parent proofs — where every symbol here is put to work.
Change of base formula — built from the same definition-plus-indices toolkit.
Solving exponential equations with logs — where the domain restrictions earn their keep.
Natural logarithm ln and e — the base- $e$ special case; identical laws.
Graphs of logarithmic functions — the "always positive" picture behind $x > 0$ .