Solving exponential equations using logarithms: Foundations

Before you can solve $2^{x} = 7$ , you must be fluent in seven small pieces of notation. The parent note fires them off quickly. Here we slow all the way down and earn each one, in an order where every piece rests on the one before it.

0. The picture the whole topic lives in

Everything below happens on one machine: pick a fixed number, called the base, and multiply it by itself some number of times. That "number of times" is the exponent — and it is what we usually don't know.

Figure — Solving exponential equations using logarithms

1. Powers: what $b^{x}$ actually means

Why the topic needs this. If you can't read $2^{x}$ as "2 multiplied by itself $x$ times", the sentence " $x$ is stuck in the exponent" means nothing. This is rung one of the ladder.

But here is the catch that makes the whole topic necessary: in $2^{x} = 7$ , the answer is not a whole number ( $2^{2} = 4$ is too small, $2^{3} = 8$ is too big). So $x$ lands between 2 and 3. We must extend the meaning of $b^{x}$ to fractions and any real number.

2. The index laws you will actually use

The parent note quietly assumes you can flatten stacked powers. Two laws do all the work, and they hold for any real exponents (this is exactly why the extension in §1 was designed the way it was).

Why the topic needs these. The power-of-a-power law is the single fact used to derive the power law of logarithms (parent note: " $(b^{y})^{k} = m^{k}$ "). The same-base law is the shortcut when both sides share a base.

3. The equals sign as a legal move

Later (once we have met the log in §5) we will add one more legal move to this list: applying the same function — such as $lo g_{b}$ — to both sides. It is the same principle: one operation, done identically to both pans, never tips the balance.

Why the topic needs this. "Take the log of both sides" is legal only because of this principle. You are applying one function to a single number written two ways — the balance never tips.

4. Functions and their inverses — the undo-button

See Inverse Functions for the full treatment. Here is the one pairing we need, pictured:

5. The logarithm symbol itself

Now the star of the show. It is nothing but the undo-button of Section 4, given a name and a notation.

Why the topic needs this. Section 4 said "logs undo exponentials". Section 5 gives that undo a symbol you can write on paper and a question you can say out loud. Every step of the parent's method is this symbol at work.

6. The power law — where every symbol combines

Now that "power", "base", "exponent", index laws, and the log symbol all exist, the parent note's key formula can be derived cleanly — using nothing but pieces already on this page.

Formula The power law, derived algebraically

Claim: $lo g_{b} (m^{k}) = k lo g_{b} (m)$ (with $b > 0, b \neq = 1, m > 0$ ).

Step 1 — rewrite $m$ using the undo-pair (§4). Because $b^{l o g_{b} m} = m$ (do-then-undo), we may replace $m$ by that expression: $m^{k} = (b^{l o g_{b} m})^{k} .$ Why: it swaps $m$ for something with the base $b$ showing, so an index law can bite.

Step 2 — flatten with power-of-a-power (§2). Using $(b^{y})^{k} = b^{y k}$ with $y = lo g_{b} m$ : $(b^{l o g_{b} m})^{k} = b^{k l o g_{b} m} .$ Why: this collapses the double exponent into a single exponent, $k lo g_{b} m$ .

Step 3 — take $lo g_{b}$ of both sides (§3 + §4). Apply the same function $lo g_{b}$ to $m^{k} = b^{k l o g_{b} m}$ : $lo g_{b} (m^{k}) = lo g_{b} (b^{k l o g_{b} m}) = k lo g_{b} m .$ Why: the right side is exactly "log undoes exp" ( $lo g_{b} (b^{stuff}) = stuff$ ). $lo g_{b} (m^{k}) = k lo g_{b} (m)$ What it does (picture): it slides the exponent $k$ down from its top-of-ladder position and turns it into an ordinary multiplier out front. That is the move that frees a trapped unknown.

For the product and quotient partners of this law, see Laws of Logarithms; for why $\frac{lo g c}{lo g b} = lo g_{b} c$ , see Change of Base Formula.

b^{x}

, what is the base and what is the exponent?

b

is the base (multiplied by itself);

x

is the exponent, the number of times.

What does

2^{4}

equal, and why is it NOT

2 \times 4

2 \times 2 \times 2 \times 2 = 16

; the exponent counts copies in a product, not a factor.

How is

b^{1/2}

defined, and why?

b^{1/2} = b

, because

(b^{1/2})^{2} = b^{1} = b

must hold by the power-of-a-power law.

Why must the base

b

be positive for

b^{x}

to work for all real

x

Negative bases give non-real values (e.g.

(- 1)^{1/2} = - 1

), so we require

b > 0

Simplify

(b^{y})^{k}

using an index law.

b^{y k}

— power of a power multiplies the exponents.

b^{p} = b^{q}

(same fixed base

b

), what can you conclude?

p = q

; equal outputs force equal exponents.

Why is "apply the same function to both sides" a legal move?

An equation is a balance; one operation done identically to both pans keeps it level.

State the two defining properties of an inverse function

f^{- 1}

f (f^{- 1} (x)) = x

and

f^{- 1} (f (x)) = x

— do-then-undo lands you back at the start.

What question does

lo g_{b} (m)

ask?

"To what power must I raise

b

to get

m

What are the domain restrictions on

lo g_{b} (m)

Base

b > 0

and

b \neq = 1

; input

m > 0

(since

b^{x}

is always positive).

Evaluate

lo g_{2} (8)

and

lo g_{10} (1000)

3

and

3

(since

2^{3} = 8

and

1 0^{3} = 1000

What does

lo g

(no base) mean, and what warning comes with it?

Usually base 10 in school/calculators, but base

e

in higher maths and much software — always check the convention.

Which operation is the inverse (undo) of

b^{x}

lo g_{b}

— the logarithm to base

b

Give the algebraic derivation of

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

m^{k} = (b^{l o g_{b} m})^{k} = b^{k l o g_{b} m}

; take

lo g_{b}

of both sides to get

k lo g_{b} m

What does

\approx

signal, and what is a safe default precision?

"Approximately equal" (rounded); round only at the end, default 3 significant figures.

Connections

Inverse Functions — why log is the exact undo of exponentiation.
Laws of Logarithms — the product and quotient partners of the power law.
Change of Base Formula — turns $lo g_{b} c$ into $\frac{l o g c}{l o g b}$ .
The Number e and Natural Logarithms — the base- $e$ log $ln$ .
Exponential Growth and Decay — where these equations show up.
Solving Equations Graphically — the intersection picture of $y = b^{x}$ and $y = c$ .

Foundations — Solving exponential equations using logarithms

0. The picture the whole topic lives in

1. Powers: what $b^{x}$ actually means

2. The index laws you will actually use

3. The equals sign as a legal move

4. Functions and their inverses — the undo-button

5. The logarithm symbol itself

6. The power law — where every symbol combines

7. The approximation sign

Prerequisite map

Equipment checklist

Connections