Lesson 1 of 4

Growth by multiplying.

About 20 minutes · 5 practice questions

Read the full article: Exponential growth and logarithms

An amount multiplied by the same number at every step: why ten doublings give more than a thousand, how to write it as a power, and why such growth always overtakes a steady addition in the end.

Step 1 of 5 · Ten doublings

Step 1 of 5

Ten doublings

In the Unit on linear functions you met amounts that grow by a steady addition: each step adds the same number, and the graph is a straight line. Here the amount grows by a steady multiplication: each step multiplies it by the same number. Growth like this is called exponential growth.

On the board, each column is the amount after one more step. The first column, at step $0$ ⁠, is the start: $1$ ⁠. Growth per step is locked at $\times 2$ ⁠, so the amount doubles at every step. Right now you see three steps: $1, 2, 4, 8$ ⁠. Predict what there will be after ten.

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Step 2 of 5

Powers

Multiplying by the same number again and again is written as a power. Four steps of $\times 3$ ⁠ are $3 \times 3 \times 3 \times 3$ ⁠, or for short $3^4$ ⁠: the base $3$ ⁠ is the number you multiply by, and the exponent $4$ ⁠ counts how many times. After $n$ ⁠ steps of $\times g$ ⁠, starting from $1$ ⁠, the amount is $g^n$ ⁠.

Count the slabs in the last column: one pale slab for the start, and one more slab for each step.

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Step 3 of 5

Each step adds more

In linear growth every step adds the same. Here every step adds a new slab, and the amber slab on top of each column is what the last step added. Compare the slabs with each other: when doubling, each new slab is exactly as big as everything beneath it.

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Step 4 of 5

Passing a thousand

$1024$ ⁠ is very close to $1000$ ⁠, so ten doublings are about a thousandfold. Twenty doublings are about a millionfold, and thirty about a billionfold. Here Growth per step is locked at $\times 3$ ⁠, faster growth still. Change Steps and find the first step at which the amount passes a thousand.

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Step 5 of 5

Steady addition or doubling

Compare two plans. In the first you start from $0$ ⁠ and add $100$ ⁠ at every step: that is the linear function $y = 100x$ ⁠, with slope $100$ ⁠. In the second you start from $1$ ⁠ and double at every step: $y = 2^x$ ⁠. The board shows the second. After six steps it holds only $64$ ⁠, while the first is already at $600$ ⁠. Move Steps and follow the two.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

The amount starts at $1$ ⁠ and doubles at every step. How much will it be after ten steps?

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Practice · Calculate

Growth per step is locked at $\times 3$ ⁠ and Steps at $3$ ⁠, so the last column is $3^3$ ⁠. How much will there be after one more step? That is, what is $3^4$ ⁠?

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Practice · Calculate

The amount doubles at every step, and there are $6$ ⁠ steps. By how much did the sixth step increase it? In other words, how big is the amber slab in the last column?

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Practice · Calculate

The amount starts at $1$ ⁠ and is multiplied by $3$ ⁠ at every step. At which step does it first pass $1000$ ⁠?

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Practice · Choose

Plan A starts from $0$ ⁠ and adds $100$ ⁠ at every step. Plan B starts from $1$ ⁠ and doubles at every step. Which is bigger after $12$ ⁠ steps?