Common log (log₁₀) and natural log (ln x): Foundations

This page builds the parent note Common log and natural log from absolutely nothing. If any line below feels obvious, good — it means the foundation is solid. We go symbol by symbol, and no symbol is used before it is drawn.

1. What is a "power" / exponent? (the seed of everything)

Look at the picture below: each rung of the ladder is one multiplication by $b$ . Climbing the ladder is exponentiation; the height you reach is $b^{n}$ .

Figure — Common log (log₁₀) and natural log (ln x)

Figure 1 — The exponent ladder for $b = 2$ : standing on the ground is $2^{0} = 1$ , and each step up multiplies the height by $2$ ( $2^{1} = 2$ , $2^{2} = 4$ , …). The rung number you stand on is exactly the exponent.

Why the topic needs this. A logarithm will ask "how many rungs did I climb?" You cannot ask that question until you can see the ladder. Every log on the parent page is secretly reading a rung-number off this ladder.

Recall Sanity check on tiny exponents

What is $b^{0}$ (for $b > 0$ )? ::: $1$ — you start at 1 and multiply zero times, so you never leave the ground. What is $b^{1}$ ? ::: $b$ — one step up the ladder. Why is $0^{0}$ excluded? ::: It gives two conflicting answers (1 from "empty product", 0 from "power of zero"), so it is left undefined.

2. Negative and fractional exponents (the ladder below the floor and between the rungs)

The ladder does not stop at the ground, and it does not only have whole-number rungs.

Figure 2 — The height $b^{y}$ (here $b = 10$ ) plotted for all real $y$ , whole and fractional. The curve hugs the horizontal axis on the left but never touches or crosses it: $b^{y} > 0$ always. The marked point shows the half-rung $1 0^{1/2} = 10 \approx 3.16$ .

3. The logarithm: reading the rung number

Now we can define the star of the topic. Exponentiation asks "I climbed $y$ rungs from base $b$ — how high am I?" The logarithm asks the reverse: "I'm at height $x$ — how many rungs did I climb?"

Figure 3 — The climbing curve $e^{x}$ (yellow) and the rung-reading curve $ln x$ (blue) are mirror images across the dashed diagonal $y = x$ . Reflecting in that diagonal swaps input and output, which is the visual meaning of "inverse function". The pair $(1, e)$ and $(e, 1)$ shows the swap.

The figure shows the two graphs mirrored across the diagonal line $y = x$ . Reflecting a graph in that diagonal is the picture of "swap input and output", which is precisely what an inverse does. That mirror image is why $lo g_{b} x$ is defined only for $x > 0$ : the exponential graph never dips to zero height, so its mirror never extends left of the vertical axis.

4. The two special bases: $10$ and $e$

The parent note singles out two rung-sizes (both satisfy $b > 0, b \neq = 1$ ).

5. The symbols that appear in the derivations (calculus alphabet)

The parent note derives $\frac{d}{d x} ln x = \frac{1}{x}$ and uses limits to build $e$ . You need three more symbols to read those lines.

Why the topic needs these. The whole reason base $e$ is called natural is a calculus fact: the slope of $e^{x}$ equals $e^{x}$ . You cannot appreciate that sentence without the slope symbol $\frac{d y}{d x}$ . See Differentiation of ln x and e^x for the full workout.

b^{n}

means in words

"Start at 1 and multiply by

b

n

times."

I know why

b^{0} = 1

and why

0^{0}

is excluded

Multiplying zero times leaves you at 1;

0^{0}

gives two conflicting answers so it is undefined.

I can rewrite

1 0^{- 3}

as a decimal

\frac{1}{1000} = 0.001

(three steps down the ladder).

I know what

b^{1/ n}

means

The

n

th root

n b

— the number that raised to the power

n

gives

b

I know why

b^{y}

is never zero or negative

You only multiply/divide/root positive numbers, so you approach but never reach 0.

I can convert

lo g_{b} x = y

into exponential form

b^{y} = x

I know the conditions the base

b

must satisfy

b > 0

and

b \neq = 1

(else the log is undefined or trivial).

I know why

lo g

needs

x > 0

No rung has negative or zero height, so no power gives

x \leq 0

I can undo an exponential with its log

lo g_{b} (b^{y}) = y

and

b^{l o g_{b} x} = x

(inverse functions).

I can read

\frac{d y}{d x}

in plain words

The slope/steepness of the graph — tiny rise over tiny run.

I can say what the

d x

\int f (x) d x

does

It names the variable and marks the tiny slice-width being summed.

I can state the power law of logs

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

— the exponent slides to the front.

Connections

Laws of Logarithms (product, quotient, power) — the three laws quoted in §6.
Exponential functions e^x and 10^x — the ladders whose rung-counts logs read.
Solving exponential equations — where the power law pulls exponents down.
Differentiation of ln x and e^x — home of the derivative symbol in §5.
Exponential growth and decay models — real-world reason $e$ is "natural."
Scientific notation and orders of magnitude — order-of-magnitude reading of §7.

Foundations — Common log (log₁₀) and natural log (ln x)

1. What is a "power" / exponent? (the seed of everything)

2. Negative and fractional exponents (the ladder below the floor and between the rungs)

3. The logarithm: reading the rung number

4. The two special bases: $10$ and $e$

5. The symbols that appear in the derivations (calculus alphabet)

6. The three log laws you must already trust

7. Order-of-magnitude thinking (why $lo g_{10}$ is friendly)

Prerequisite map

Equipment checklist

Connections