Solving logarithmic equations: Foundations — Maths deep dive

This page assumes nothing. If the parent note used a symbol, we build it here from the ground up, in an order where each idea leans only on the ones before it.

0. The starting picture: repeated multiplication

Before any fancy symbol, hold one image in your head: a number being multiplied by itself over and over.

Look at the red staircase. We start at $1$ and each step multiplies by $2$ : $1 \to 2 \to 4 \to 8 \to 16$ . The horizontal axis counts how many times we multiplied (the "power"), the vertical axis shows the result.

Two questions live in this one picture, and they are opposites:

Going right (multiply): "I took $3$ steps — where do I land?" Answer: $8$ .
Going backward (detective): "I landed on $8$ — how many steps was that?" Answer: $3$ .

The first question is exponentiation. The second is the logarithm. Keep this picture; every symbol below is a label on some part of it.

1. The exponent (index / power)

Picture: $b$ is the size of each step on the staircase; $n$ is the number of steps you take. So $2^{3} = 2 \times 2 \times 2 = 8$ is "step size $2$ , taken $3$ times, lands on $8$ ."

Why the topic needs it: every log is defined in terms of an exponent. You cannot flip a sentence you cannot first write forwards.

Why these three are true (not memorised): $b^{x} \cdot b^{y}$ means " $x$ copies of $b$ , then $y$ more copies" — that is $x + y$ copies in total. Dividing removes copies (subtract). A power of a power repeats the multiplication $k$ times over (multiply). These are the only facts the parent's log laws stand on.

2. The logarithm — the "power detective"

So $lo g_{2} (8) = 3$ because $2^{3} = 8$ . Read it out loud as a question: " $2$ to what gives $8$ ?"

Picture: on the staircase, $lo g_{2} (8)$ is you reading the horizontal axis after landing at height $8$ — you point at the " $3$ " underneath. Exponentiation reads the vertical axis; the log reads the horizontal one.

3. The three named parts of a log

Every $lo g_{b} (N)$ has three roles. Losing track of which is which is where most errors start.

Why the topic needs this split: the parent's three strategies each act on a different part. "Exponentiate" attacks the whole equation; "combine logs" works inside the argument; "equate arguments" throws away the log once two arguments face off. You must know which part you are touching.

4. Why the argument must be positive (the domain rule)

This is the single most-tested trap in the whole topic, so we earn it with a picture, not a rule.

Look at the red exponential curve $y = b^{x}$ (with $b > 1$ ). Every output — every height — is strictly above zero. No matter how far left you go, the curve approaches the floor $y = 0$ but never touches it, and never dips below. So a positive base can only ever produce positive numbers.

Now the log is this curve read backwards (its inverse). Since forwards it never produced $0$ or a negative, backwards it can never accept $0$ or a negative as its argument. Asking $lo g_{2} (- 8)$ is asking " $2$ to what power gives $- 8$ ?" — and the answer is nothing exists.

Why $b \neq = 1$ : $1$ to any power is still $1$ , so the staircase never moves — the detective could never distinguish the powers. A useless base.

Why the topic needs it: algebra can hand you a number that makes an argument negative. That number was never a real solution — it is an extraneous root. This is exactly why the parent insists you always check. See Domain and range of log functions for the full story.

5. Log and exponent are inverse functions

Adding $5$ then subtracting $5$ returns your number. In the same way: $b^{l o g_{b} N} = N and lo g_{b} (b^{x}) = x$

Picture: walk right along the staircase (exponentiate), then read the axis backwards (log) — you are back at your starting step. See Inverse functions.

Why the topic needs it: Strategy 1 ("isolate and exponentiate") is literally applying this cancellation. Raising $b$ to both sides of $lo g_{b} N = c$ turns the left side into plain $N$ .

b^{n}

mean in plain words?

Multiply the base

b

by itself

n

times;

b

is the step size,

n

the number of steps.

Rewrite

lo g_{b} (a) = c

without a log.

b^{c} = a

— the same fact facing the other way.

lo g_{5} (x - 4)

, which part is the argument?

x - 4

— the number inside the log.

Why can't

lo g_{2} (- 8)

exist?

A positive base only ever produces positive numbers, so no power of

2

gives

- 8

Why must the base satisfy

b \neq = 1

1

to any power is

1

, so it can never distinguish different powers.

State the index law for

b^{x} \cdot b^{y}

b^{x + y}

— adding copies.

What does it mean that log and exponent are inverses?

One undoes the other:

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

and

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

Why may we conclude

M = N

from

lo g_{b} M = lo g_{b} N

The log is one-to-one, so equal outputs force equal inputs (with

M, N > 0

What is an extraneous root?

A number the algebra produces that makes a log argument

\leq 0

, so it isn't a real solution — reject it.

When may you NOT equate arguments?

When one side is a plain number, not a log; combine into one log and exponentiate instead.

Connections

3.2.11 Solving logarithmic equations (Hinglish)
Laws of logarithms
Exponential equations
Change of base formula
Inverse functions
Quadratic equations by substitution
Domain and range of log functions

Foundations — Solving logarithmic equations

0. The starting picture: repeated multiplication

1. The exponent (index / power)

2. The logarithm — the "power detective"

3. The three named parts of a log

4. Why the argument must be positive (the domain rule)

5. Log and exponent are inverse functions

6. One-to-one: why we may "equate arguments"

7. Notation you'll meet in passing

Prerequisite map

Equipment checklist

Connections