🔊Logarithms

Multiplication has division at its back, squaring has square roots for company — and exponents? Their inverse is the logarithm. $2^3 = 8$; asked backwards, "2 to what power gives 8?", the answer 3 is written $\log_2 8 = 3$. Do not let the new symbol scare you: it is shorthand for one exponential question.

Rewinding a power

More formally: if $a^c = b$, we write $\log_a b = c$. The left side asks "how many powers of a make b", and the right side is the answer. Both spellings say the same thing:

$2^5 = 32 \quad \Longleftrightarrow \quad \log_2 32 = 5$

$10^3 = 1000 \quad \Longleftrightarrow \quad \lg 1000 = 3$

Checking is always the same move: put c back into the exponent and see whether b returns. This rewind is the same idea as a square root undoing a square — inverse operations exist to run the film backwards.

Power machine, logarithm machine

A secret rule hides inside the machine. Feed it numbers and guess the rule!

▶ Input

⚙️? ? ?

Output

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🔍 Reveal rule🎲 New rule

A rule hides inside the machine: feed it an input, watch the output, and guess the rule. After playing, think backwards — if the rule is $x^2$, the machine that undoes it is the square root; if the rule is $2^x$, the undo-machine turns 8 back into 3 and 32 back into 5, and that rewinding machine is the logarithm. Inverse machines swap input and output. The formal name for such a machine is a function — they get far more stage time there.

Two laws of logarithms

The most likeable talent of logarithms: they demote hard operations by one level.

In exponent world	In logarithm world	Numeric check
Multiplying	becomes adding	$\lg 100 + \lg 1000 = 2 + 3 = 5 = \lg 100000$ ✓
Powers	become coefficients	$\log_2 8 = \log_2 2^3 = 3\log_2 2 = 3$ ✓

Multiply-becomes-add is one glance away: $100 = 10^2$ and $1000 = 10^3$, and multiplying the numbers adds the exponents, $2 + 3 = 5$ — so the logarithms add too. Power-becomes-coefficient is just as handy: 8 already is $2^3$, and the logarithm simply lifts the exponent 3 down.

A ruler for orders of magnitude

Logarithms were born for numbers that leap across many zeros. Scientific notation squeezes big numbers into powers of ten; the logarithm goes further — it reads the exponent straight off as a tick mark: every ×10 in value moves the scale one step.

• The Richter scale for earthquakes is such a ruler: one step up in magnitude means ten times the ground amplitude. Magnitudes 6 and 4 differ by 2 steps, so the amplitude differs $10^2 = 100$ times.
• The pH of a solution works the same way: it is defined as $-\lg c$ (c the hydrogen ion concentration). pH 3 sits 3 steps below pH 6, so its concentration is $10^3 = 1000$ times larger — far more acidic.

Seat numbers on a logarithmic ruler, and wildly different sizes fit into one picture.

Check yourself

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Quick quiz

↺ Try again

1. 1. What is log₂32?

   456

2. 2. log₂4 + log₂8 equals log₂ of what?

   log₂12log₂32log₂6

3. 3. Solution A has pH 3 and solution B has pH 6. A's hydrogen ion concentration is how many times that of B?

   2 times100 times1000 times