Logarithms.

Step 1 of 6

How many times?

A power answers the question “what do you get if you multiply $2$⁠ by itself three times?”: $2^3 = 8$⁠. A logarithm asks the opposite question: “how many times must you multiply $2$⁠ by itself to reach $8$⁠?” You write $\log_2 8$⁠ and read it “the logarithm of $8$⁠ to base $2$⁠”. The base is the number you multiply by, and the answer is the exponent.

So $\log_2 8 = 3$⁠, because $2^3 = 8$⁠.

Growth per step is locked at doubling. The number of steps on the board is exactly the logarithm to base $2$⁠ of the value above the last column.

Step 2 of 6

The log scale

On an ordinary scale, equal heights are equal amounts: the distance from $0$⁠ to $100$⁠ equals the distance from $900$⁠ to $1000$⁠. On a log scale, equal heights are equal multiples: the distance from $1$⁠ to $10$⁠ equals the distance from $10$⁠ to $100$⁠ and from $100$⁠ to $1000$⁠. The height of a point on such a scale is the logarithm of its value. Predict how the doubling curve will look when you choose Log scale under Look inside.

Step 3 of 6

Powers of ten

The common logarithm is to base $10$⁠: $\log_{10} 100 = 2$⁠, because $10^2 = 100$⁠. It counts the digits of a number, less one: $\log_{10}$⁠ of a four-digit number is between $3$⁠ and $4$⁠. On the board, the lines for $1, 10, 100, 1000$⁠ are equally far apart. Switch Look inside between Ordinary scale and Log scale and watch them spread out.

Step 4 of 6

Every step climbs the same height

On the log scale all the slabs are the same height, because every step multiplies by the same factor. The board shows how many powers of ten each step climbs: $\log_{10} 3 \approx 0.477$$\log_{10} 3 \approx$$0.477$⁠. Four steps climb four times as much, so $\log_{10} 3^4 = 4 \cdot \log_{10} 3$$\log_{10} 3^4 =$$4 \cdot \log_{10} 3$⁠. That is the rule: the logarithm of a power is the exponent times the logarithm of the base.

Step 5 of 6

The logarithm of a product

Two slabs stacked on each other are two multiplications in a row, and on the log scale their heights add up. That is the product rule, the reason logarithms were invented: they turn multiplication into addition.

$\log(a \cdot b) = \log a + \log b$$\log(a \cdot b) =$$\log a + \log b$⁠

Change Growth per step and Steps, and check that each step climbs the same height and that the heights add up.

Step 6 of 6

Solving with a logarithm

With the rule you can answer “how many times?” questions that are out of reach by counting. A million is $10^6$⁠, that is $6$⁠ powers of ten, and each doubling climbs $\log_{10} 2 \approx 0.301$$\log_{10} 2 \approx$$0.301$⁠ of them. The board only goes up to $12$⁠ steps, but the arithmetic carries on.