Introduction to Logarithmic Laws

Logarithms are the inverse operation of exponentiation, and their real power comes from a set of laws that let us break apart and reassemble complex expressions. These laws are not arbitrary rules — they follow directly from the laws of exponents you already know.

Logarithm

If $a^{x} = y$ , then $lo g_{a} (y) = x$ . The logarithm answers the question: to what power must we raise $a$ to get $y$ ?

In AHL 1.9, the base $a$ is restricted to either 10 or $e$ in IB examinations:

Common logarithm: $lo g (x) = lo g_{10} (x)$ — written without a base
Natural logarithm: $ln (x) = lo g_{e} (x)$ — uses Euler's number $e \approx 2.718$

Note

For all logarithmic laws to apply, the arguments must be strictly positive (you cannot take the logarithm of zero or a negative number), and the base must be positive and not equal to 1.

There are three fundamental laws to master: the product rule, the quotient rule, and the power rule. Together they allow you to expand, condense, and simplify logarithmic expressions.

The Product Rule

Product Rule for Logarithms

For any valid base $a$ and positive numbers $x$ and $y$ :
$lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)$
The logarithm of a product equals the sum of the logarithms.

Why does this work? Let $lo g_{a} (x) = p$ and $lo g_{a} (y) = q$ , so $x = a^{p}$ and $y = a^{q}$ . Then:
$x y = a^{p} \cdot a^{q} = a^{p + q}$
$∴ lo g_{a} (x y) = p + q = lo g_{a} (x) + lo g_{a} (y) ✓$

This derivation shows the law is simply the exponent addition rule in disguise.

Example

Simplify $lo g_{10} (300)$ without a calculator.

Write $300 = 3 \times 100$ :
$lo g (300) = lo g (3 \times 100) = lo g (3) + lo g (100) = lo g (3) + 2$

Since $lo g (3) \approx 0.477$ , we get $lo g (300) \approx 2.477$ . ✓ (Check: $1 0^{2.477} \approx 300$ )

Example

Given $ln (2) \approx 0.693$ , find $ln (8)$ .

$ln (8) = ln (2 \times 4) = ln (2) + ln (4) = ln (2) + ln (2 \times 2) = ln (2) + ln (2) + ln (2) = 3 ln (2) \approx 2.079$

(Note: this also follows from the power rule — see below.)

Warning

The product rule only applies when all logarithms share the same base. You cannot combine $lo g (x) + ln (y)$ using this rule.

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AHL 1.9—Log laws

Introduction to Logarithmic Laws

The Product Rule

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