Introduction to logarithms (base 10 and e)

IB Mathematics: Applications and Interpretation SL· SL 1.5· 18 min read

1. What Is a Logarithm?★★☆☆☆⏱ 6 min

A logarithm answers the question: to what power must we raise a base to get a given number? If $a^{x} = b$ , then $x$ is the logarithm of $b$ to base $a$ . So a logarithm is simply the inverse operation of raising a base to a power.

📘 Definition

Definition of a Logarithm

\log_a b = x \iff a^x = b

For a base $a > 0$ , $a \neq = 1$ , and argument $b > 0$ : $lo g_{a} b = x$ means exactly that $a^{x} = b$ . The base of the log matches the base of the power, and the log gives the exponent.

Example:

$lo g_{2} 8 = 3$ because $2^{3} = 8$

This equivalence lets you rewrite any exponential statement as a logarithm and vice versa. Reading the relationship both ways is the single most useful skill for this topic.

📐 Worked Example

Rewrite $1 0^{3} = 1000$ in logarithmic form, and rewrite $lo g_{5} 25 = 2$ in exponential form.

1
For $1 0^{3} = 1000$ , the base is 10, the exponent is 3, and the result is 1000. Using $a^{x} = b \Leftrightarrow lo g_{a} b = x$ :
2
$10^3 = 1000 \iff \log_{10} 1000 = 3$
3
For $lo g_{5} 25 = 2$ , the base is 5, the log value (exponent) is 2, and the argument is 25:
4
$\log_5 25 = 2 \iff 5^2 = 25$

Exam tip:

If you are stuck on a logarithm, immediately rewrite it as a power: $lo g_{a} b = x$ becomes $a^{x} = b$ . This turns an unfamiliar log into a familiar exponent question.

2. Common and Natural Logarithms★★☆☆☆⏱ 5 min

Two bases are used so often that they have their own shorthand. Base 10 gives the common logarithm, written $lo g b$ (sometimes $l g b$ ). Base $e$ (where $e \approx 2.718$ ) gives the natural logarithm, written $ln b$ .

📘 Definition

Common and Natural Logarithms

Common logarithm: $lo g b = lo g_{10} b$ , so $lo g b = x$ means $1 0^{x} = b$ . \ Natural logarithm: $ln b = lo g_{e} b$ , so $ln b = x$ means $e^{x} = b$ .

Example:

$lo g 100 = 2$ because $1 0^{2} = 100$ ; $ln e = 1$ because $e^{1} = e$

Natural logarithms appear throughout IB AI SL when modelling exponential growth and decay, because continuous growth is naturally described using base $e$ . Common logarithms appear in applied scales such as pH, sound intensity, and the Richter scale.

3. Evaluating Logarithms with Technology★★☆☆☆⏱ 6 min

Some logarithms give exact whole-number values that you can find by inspection, such as $lo g 1000 = 3$ . Most logarithms, however, are not whole numbers and are evaluated numerically using your GDC.

For a base-10 logarithm, use the $lo g$ key: e.g. $lo g 250 \approx 2.40$ .
For a natural logarithm, use the $ln$ key: e.g. $ln 12 \approx 2.48$ .
Most GDCs also have a general $lo g_{a} b$ template that lets you enter any base directly.

📐 Worked Example

Use a GDC to evaluate $lo g 45$ and $ln 20$ , each to 3 significant figures.

1
Enter $lo g 45$ using the log key (base 10):
2
$\log 45 = 1.6532\ldots \approx 1.65$
3
Enter $ln 20$ using the ln key (base $e$ ):
4
$\ln 20 = 2.9957\ldots \approx 3.00$
5
Check each answer makes sense: $1 0^{1.65} \approx 45$ and $e^{3.00} \approx 20$ , both consistent.

✓ Quick check

Test your understanding:

What is the value of $lo g_{4} 16$ ?
- 2
- 4
- 8
- 64
Reveal answer
2 —
Correct! $lo g_{4} 16 = 2$ because $4^{2} = 16$ . Rewriting the log as a power makes this quick.

4. Common Pitfalls

Wrong move:

Reading $lo g_{a} b$ as $a^{b}$

Why:

A logarithm returns the exponent, not the power. $lo g_{a} b$ is the number $x$ such that $a^{x} = b$ .

Correct move:

Use $lo g_{a} b = x \Leftrightarrow a^{x} = b$ : for example $lo g_{2} 8 = 3$ , not $2^{8}$ .

Wrong move:

Swapping the base and the argument, e.g. writing $lo g_{8} 2 = 3$

Why:

The base sits at the bottom and the argument is inside. $lo g_{2} 8 = 3$ , but $lo g_{8} 2 = \frac{1}{3}$ .

Correct move:

Keep the base as the number being raised to a power: $2^{3} = 8$ gives $lo g_{2} 8 = 3$ .

Wrong move:

Taking the logarithm of a negative number or zero, e.g. $lo g (- 5)$

Why:

A positive base raised to any real power is always positive, so a log of a non-positive number does not exist.

Correct move:

State that $lo g b$ is only defined for $b > 0$ .

Wrong move:

Assuming the $lo g$ key on a GDC means natural log

Why:

On most calculators the $lo g$ key is base 10 and the separate $ln$ key is base $e$ ; mixing them gives wrong values.

Correct move:

Use $lo g$ for base 10 and $ln$ for base $e$ , or the general $lo g_{a} b$ template for any base.

5. Quick Reference Cheatsheet

Idea	Statement	Notes
Definition	$lo g_{a} b = x ⟺ a^{x} = b$	$a > 0, a \neq = 1, b > 0$
Common log	$lo g b = lo g_{10} b$	Use the $lo g$ key
Natural log	$ln b = lo g_{e} b$	Use the $ln$ key, $e \approx 2.718$
Log of the base	$lo g_{a} a = 1$	Since $a^{1} = a$
Log of 1	$lo g_{a} 1 = 0$	Since $a^{0} = 1$
Valid argument	$b > 0$	Log undefined for $b \leq 0$

What's Next

Understanding logarithms as the inverse of exponentials is the foundation for solving exponential equations, where you take a logarithm of both sides to bring an unknown down from the exponent. This skill is heavily tested across both IB AI SL papers and appears in applied contexts such as exponential growth and decay, compound interest, and half-life problems.