A line adds the same amount every step. An exponential multiplies by the same amount, and that is why it explodes. A logarithm counts how many times you multiplied. Drag the growth factor and watch the curve pull away from the line.
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| step x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| curve | 1 | 1.5 | 2.25 | 3.38 | 5.06 |
| line | 1 | 1.5 | 2 | 2.5 | 3 |
In a table, a line adds the same amount each step: 1, 2, 3, 4, 5. An exponential multiplies by the same amount: 1, 2, 4, 8, 16. Calling every rising curve a line is the first mistake.
At first the two look alike. Then the multiplying one runs away, because each step multiplies a bigger number than the last.
A line adds the same amount each step. An exponential multiplies by the same amount.
To grow by 10%, multiply by 1.1, not by 0.1. The growth factor is 1 plus the rate. To shrink by 10%, multiply by 0.9.
Start at 100 and apply it three times: 110, 121, 133.1. Each step is 10% of a bigger number, so the amount added grows too.
10% more is times 1.1: the factor is 1 plus the rate.
When the factor is between 0 and 1 the curve decays: it shrinks every step. It heads toward the x-axis but never reaches it, and never goes below zero.
That floor is a horizontal asymptote. Halving 2 gives 1, 0.5, 0.25 and so on, always smaller and always above 0.
The curve creeps along the ground and never touches it.
log₂ 8 asks: how many times do I multiply 1 by 2 to reach 8? The answer is 3.
A logarithm and an exponent say the same fact two ways: 2³ = 8 and log₂ 8 = 3. To solve 2ˣ = 20, take the log of both sides: x = log 20 ÷ log 2, about 4.32.
A logarithm counts how many times you multiplied.
Linear growth adds the same amount each step. Exponential growth multiplies by the same factor each step. Exponential growth starts similar but ends up far larger.
Add the rate to 1. A 10% increase has a factor of 1.1, a 5% increase 1.05. For a decrease subtract: 10% less is a factor of 0.9.
The answer to the question “what power gives this number?”. Since 2³ = 8, log₂ 8 = 3. It counts how many times you multiplied.
Take the logarithm of both sides. For 2ˣ = 20, x = log 20 ÷ log 2, which is about 4.32. Taking the log of only one side is the usual mistake.
World 7 of Pip: Math Adventure is about exponentials and logs. You grow a vine to the clouds, and learn how long growth and decay take.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.