Visual walkthrough — Laws of logarithms — product, quotient, power rules — proofs

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Step 0 — What even IS a logarithm? (the one idea everything rests on)

WHAT. Pick a fixed number to multiply by, over and over. Call it the base, written $b$. Start at $1$ and keep multiplying by $b$. A logarithm just counts how many times you multiplied.

$\underset{\text{"how many }b\text{’s?"}}{\underbrace{{\log }_b}}\underset{\text{the number you landed on}}{\underbrace{x}}=\underset{\text{the count}}{\underbrace{y}}⟺\underset{\text{multiply }b\text{, }y\text{ times}}{\underbrace{b^y}}=\underset{\text{result}}{\underbrace{x}}$

• $b$ — the base, the thing we repeatedly multiply by (we need $b>0,b\ne 1$).
• $x$ — the number we ended up at (must be $x>0$).
• $y$ — the answer: the number of times we multiplied. This is the log.

WHY this and not just "divide"? Because repeated multiplication doesn't undo with division — it undoes with counting the multiplications. The log is the tool built specifically to answer the question "$b$ to what power gives $x$?" Nothing else answers that.

PICTURE. Below, the base is $b=2$. Each step to the right is one multiplication by $2$. The red number under each landing spot is its logarithm — literally the step count.

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Step 1 — Name the two logs so we can hold them

WHAT. We want to understand ${\log }_b(MN)$ — the log of a product of two positive numbers $M$ and $N$. First, give each separate log a short name:

$\underset{\text{name for }M\text{’s step-count}}{\underbrace{p}}={\log }_{b}M,\underset{\text{name for }N\text{’s step-count}}{\underbrace{q}}={\log }_{b}N$

• $M,N$ — two positive numbers we will multiply.
• $p$ — how many times you multiply $b$ to reach $M$.
• $q$ — how many times you multiply $b$ to reach $N$.

WHY name them? We can't manipulate an unknown we keep spelling out in full. Naming $p$ and $q$ turns vague "logs" into ordinary numbers we can add later. This is the single move that makes the whole proof possible.

PICTURE. Two separate step-counts on the same number line: $p$ steps land on $M$, $q$ steps land on $N$.

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Step 2 — Translate each name back into "multiply by $b$"

WHAT. The definition from Step 0 says a log is an exponent, so we rewrite $p$ and $q$ in exponential form:

$b^p=M\text{and}{b}^q=N$

• $b^p=M$ — "$p$ multiplications by $b$ lands you exactly on $M$."
• $b^q=N$ — "$q$ multiplications by $b$ lands you exactly on $N$."

WHY undo the log? Products of powers of the same base have a beautiful, simple rule (next step). Logs by themselves don't multiply nicely — but powers of $b$ do. So we leave "log-land" and enter "exponent-land" where the algebra is easy.

PICTURE. The same two landings, now labelled as heights $b^p$ and $b^q$ — this is the identical picture as Step 1, just relabelled to remind us $M$ and $N$ are secretly powers of $b$.

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Step 3 — Multiply, and watch the exponents ADD

WHAT. Now multiply the two numbers together:

$MN=b^p\cdot b^q=b^{p+q}$

• $MN$ — the product we actually care about.
• $b^p\cdot b^q$ — substitute what $M$ and $N$ really are.
• $b^{p+q}$ — because multiplying same-base powers adds the exponents.

WHY do exponents add? Multiplying by $b$ a further $q$ times, after already multiplying $p$ times, is just multiplying $p+q$ times in total. Counting doesn't care where you paused. This single fact — see Exponentials — laws of indices — is the engine of the product rule.

PICTURE. Stack the two step-counts end to end: $p$ steps, then $q$ more steps, reach $MN$ in $p+q$ steps total. The red bracket shows the combined count.

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Step 4 — Read the total count as a single log

WHAT. Apply ${\log }_b$ to both sides. The log of "$b$ to a power" just hands back the power (that's the counter reading off its own count):

${\log }_b(MN)={\log }_b\left(b^{p+q}\right)=p+q$

• ${\log }_b(MN)$ — the thing we set out to find.
• ${\log }_b(b^{p+q})$ — same value, written with the exponent exposed.
• $p+q$ — the log cancels the base cleanly, leaving just the exponent, because ${\log }_b(b^k)=k$.

WHY does ${\log }_b(b^k)=k$? By definition ${\log }_b$ asks "how many $b$'s?" and $b^k$ is made of exactly $k$ of them. Question and answer are already staring at each other.

PICTURE. The red bracket of total length $p+q$ from Step 4, now read as one number — the log of the landing point $MN$.

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Step 5 — Swap the names back

WHAT. Recall from Step 1 that $p={\log }_{b}M$ and $q={\log }_{b}N$. Substitute them back:

$\boxed{{\log }_b(MN)={\log }_{b}M+{\log }_{b}N}■$

• Left side — one log of a product.
• Right side — a sum of two separate logs.

WHY this is the finish. Every symbol we invented ($p$, $q$) has been paid back. What remains is a statement about the original quantities only — a genuine law.

PICTURE. Product on the left, sum on the right, joined by the red equals sign — the mirror that turns $\times$ into $+$.

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Step 6 — The edge cases (never leave a gap)

WHAT. Check the boundaries where a careless reader might trip.

Case $N=1$. Then $MN=M$, and ${\log }_{b}1=0$ (zero steps: $b^0=1$). The rule reads ${\log }_{b}M={\log }_{b}M+0$ ✅ — consistent.

Case $M=N$. Then $MN=M^2$ and the rule gives ${\log }_{b}M+{\log }_{b}M=2{\log }_{b}M$ — which is exactly the power rule for $k=2$. The product rule contains it.

Forbidden case $M\le 0$ or $N\le 0$. There is no count $p$ with $b^p$ equal to a non-positive number: a positive base to any real power stays positive. So ${\log }_{b}M$ simply doesn't exist, and the rule doesn't apply — you may only combine logs whose arguments are $>0$.

WHY show these. A law you can't state the limits of is a law you'll misuse. The domain restriction $M,N>0$ is not decoration — it's why the picture (a number line of positive landing points) never crosses zero.

PICTURE. The positive number line with a red "forbidden zone" at and below $0$ — no landing spot lives there.

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The one-picture summary

Everything above, compressed: multiply along the number line ⇒ add the step-counts. That is the product rule, seen in a single image.

Recall Feynman retelling — say it in plain words

A logarithm is just a step counter: how many times did I multiply by the base $b$ to get here? Take two numbers, $M$ and $N$. It took me $p$ steps to reach $M$ and $q$ steps to reach $N$. Now I multiply them together. Multiplying is continuing to walk — I take the $p$ steps to $M$, then $q$ more steps, landing on $MN$. Total steps? Just $p+q$. But "total steps to reach $MN$" is exactly ${\log }_b(MN)$, and $p+q$ is exactly ${\log }_{b}M+{\log }_{b}N$. So the log of a product is the sum of the logs — because walking further just adds to your step count. It only works while $M$ and $N$ stay positive, since you can never step onto a zero or negative landing spot.

Recall Which step is the real "engine"?

Step 3: $b^p\cdot b^q=b^{p+q}$ — same-base multiplication adds exponents. Everything else is naming and translating.

Recall Why must

$M,N>0$? $b^p$ is a positive base to a real power, always positive. Non-positive numbers have no step-count, so no log — the derivation's number line never reaches them.

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Connections

• Exponentials — laws of indices — the rule $b^p{b}^q=b^{p+q}$ that powers Step 3.
• Change of base formula — proved by the same "name-then-translate" move.
• Solving exponential equations with logs — where combining logs earns marks.
• Natural logarithm ln and e — identical picture, base $e$.
• Graphs of logarithmic functions — the visual domain $x>0$ from Step 6.