Why Scale with Logarithms?

In many real-world contexts , populations, earthquake magnitudes, sound intensity, financial data , numbers span enormous ranges. Plotting a dataset where values range from 1 to 1,000,000 on a linear scale compresses all the small values into an indistinguishable cluster near the origin.

Logarithms solve this problem by converting multiplicative scales into additive ones. Instead of working with the raw numbers, we work with their exponents.

Logarithm: The logarithm of a number $x$ to base $a$ is the exponent $n$ to which $a$ must be raised to produce $x$ . Written formally:
$lo g_{a} (x) = n ⟺ x = a^{n}$

This single idea has a powerful consequence for large numbers:

Original Value	$lo g_{10}$ Value
1	0
1,000	3
1,000,000	6
0.001	−3

The values in the table span nine orders of magnitude (from $1 0^{- 3}$ to $1 0^{6}$ ), yet the log scale compresses them into a range of just 9 units. If we consider only the positive values 1 to 1,000,000 (six orders of magnitude), those are compressed into a range of just 6 units. That's the power of logarithmic scaling.

Analogy

Think of the Richter scale for earthquakes. An earthquake of magnitude 7 isn't slightly stronger than magnitude 6 , it's 10 times more intense. The log scale converts that multiplicative jump into a neat additive step of 1.

Key Logarithm Properties

Before applying logarithms to data, you need to be fluent with their algebraic properties. These rules are the engine behind every linearisation technique.

For any valid base $a$ (where $a > 0$ , $a \neq = 1$ ):

$lo g_{a} (1) = 0$ since $a^{0} = 1$
$lo g_{a} (a) = 1$ since $a^{1} = a$
$lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)$ , Product Rule
$lo g_{a} (\frac{x}{y}) = lo g_{a} (x) - lo g_{a} (y)$ , Quotient Rule
$lo g_{a} (x^{b}) = b lo g_{a} (x)$ , Power Rule
$lo g_{a} (x) = \frac{lo g _{b} ( x )}{lo g _{b} ( a )}$ , Change of Base Formula

Exam Tip

The power rule is the most important for linearisation. When you take $lo g$ of both sides of $y = a x^{b}$ , it turns the exponent $b$ into a coefficient , that's what creates the straight line on a log-log graph.

Warning

$lo g_{a} (x + y) \neq = lo g_{a} (x) + lo g_{a} (y)$ . The product rule applies to multiplication inside the log, not addition. This is one of the most common algebraic errors students make.

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Original Value

lo g_{10}

Value

1,000

1,000,000

0.001

−3

AHL 2.10—Scaling large numbers, log-log graphs

Why Scale with Logarithms?

Key Logarithm Properties

10 more sections in this topic

AHL 2.10—Scaling large numbers, log-log graphs

Why Scale with Logarithms?

Key Logarithm Properties

10 more sections in this topic