F3PureCore

Logarithms

Q: What does a logarithm actually mean?

It means "the exponent". _a b asks what power you raise a to, to get b — it is the inverse operation of exponentiation, just as square-rooting is the inverse of squaring.

Q: Why can't you take the log of a negative number or zero?

Because a^x (for a positive base a) is always strictly positive, no exponent x can ever produce a result of zero or a negative number.

Q: What is the difference between log and ln?

x conventionally means base 10; x specifically means base e (the natural logarithm). Both work the same way — only the base differs.

If $2^{x} = 8$ , spotting that $x = 3$ is easy by inspection. But if $2^{x} = 10$ , there is no neat integer answer — you need a tool that undoes the power and hands back the exponent directly. That tool is the logarithm.

28 min Video by TLMaths Exponentials and Logarithms

The big picture

Every exponential equation $a^{x} = b$ is really asking "what power of $a$ gives $b$ ?" — and a $logarithm$ is simply the name for that power. Once exponentials and logarithms are recognised as inverse operations of each other (exactly like squaring and square-rooting), you gain the ability to solve for an exponent directly instead of guessing, which is essential the moment a growth or decay model needs a specific time or value found.

A-Level Maths F3-01: Introducing LogarithmsWatch the walkthrough, then work through the notes below.

Open on YouTube

What you'll be able to do

Understand $lo g_{a} b$ as "the power you raise $a$ to, to get $b$ "
Convert fluently between exponential form $a^{x} = b$ and logarithmic form $lo g_{a} b = x$
Evaluate simple logarithms by inspection
Recognise $lo g$ (base 10) and $ln$ (base $e$ ) notation

Defining the logarithm

The statement $lo g_{a} b = x$ means exactly the same thing as $a^{x} = b$ — it asks "what power of $a$ gives $b$ ?" and the logarithm IS that power. Here $a$ is the $base$ (must be positive, $a \neq = 1$ ) and $b$ must be positive (since $a^{x} > 0$ always, the equation $a^{x} = b$ has no solution for negative or zero $b$ ).

lo g_{a} b = x ⟺ a^{x} = b

Exponential form and logarithmic form say the same thing.

1The base is 2, the exponent 5, the result 32.

2In log form, the exponent becomes the subject:

lo g_{2} 32 = 5

Answer

lo g_{2} 32 = 5

Tip — Always identify the base first, then ask "what power gives the target number?" — most straightforward logarithms can be evaluated this way without a calculator.

Notation: log and ln

When no base is written, $lo g x$ conventionally means $lo g_{10} x$ (base 10, sometimes called the "common logarithm"). The natural logarithm $ln x$ means $lo g_{e} x$ (base $e$ , from the previous lesson) — it has its own button on a scientific calculator because it appears constantly in calculus.

1Convert to exponential form:

x = 1 0^{3}

Answer

x = 1000

Logarithms of numbers less than 1

Because $a^{- n} = \frac{1}{a ^{n}}$ , logarithms of fractions between 0 and 1 are negative. This is consistent: a negative power of a base greater than 1 gives a small positive result.

\frac{1}{8} = 2^{- 3}

2So

lo g_{2} (\frac{1}{8}) = - 3

Answer

- 3

Tip — A logarithm is negative exactly when the argument is between 0 and 1 (for a base greater than 1) — a useful sanity check on your answer.

Think like an examiner

Convert between exponential and log form as your first move on any unfamiliar-looking log question — it usually reveals a simple power immediately.
Remember the argument of a logarithm must be strictly positive — $lo g_{a} (0)$ and $lo g_{a} (negative)$ are both undefined.
Do not confuse $lo g x$ (base 10 by convention) with $ln x$ (base $e$ ) — read the question carefully.

Common misconceptions

Thinking

lo g_{a} b

means

a

multiplied by

b

in some way.

lo g_{a} b

is the EXPONENT you need on base

a

to reach

b

— a completely different operation from multiplication.

Believing

lo g_{a} (0)

equals 0.

There is no power of a positive base that gives 0, so

lo g_{a} (0)

is undefined.

Assuming

lo g x

with no base means

lo g_{e} x

By convention

lo g x

(no base shown) means base 10; the base-

e

logarithm is written

ln x

Logarithm definition

lo g_{a} b = x ⟺ a^{x} = b

The core definition.

lo g x \equiv lo g_{10} x

Common logarithm (base 10).

ln x \equiv lo g_{e} x

Natural logarithm (base e).

Stretch yourself

Without a calculator, evaluate $lo g_{4} 2 + lo g_{4} 32$ , giving your answer as a single exact value.

Hint — Express both 2 and 32 as powers of 4 — 2 is a fractional power, 32 is not a whole-number power of 4 either, so think about writing everything with base 2 first, then convert.

Questions students ask

What does a logarithm actually mean?

It means "the exponent".

lo g_{a} b

asks what power you raise

a

to, to get

b

— it is the inverse operation of exponentiation, just as square-rooting is the inverse of squaring.

Why can't you take the log of a negative number or zero?

Because

a^{x}

(for a positive base

a

) is always strictly positive, no exponent

x

can ever produce a result of zero or a negative number.

What is the difference between log and ln?

lo g x

conventionally means base 10;

ln x

specifically means base

e

(the natural logarithm). Both work the same way — only the base differs.

Key takeaways

$lo g_{a} b = x$ means exactly the same as $a^{x} = b$ — the logarithm IS the exponent.
$lo g x$ (no base) means base 10; $ln x$ means base $e$ .
The argument of a logarithm must be strictly positive.
Logarithms of numbers between 0 and 1 are negative (for a base greater than 1).

How this fits the course

Build on

The Number e and eˣ

Laws of Logarithms

Leads to

Laws of Logarithms

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