Monday
Monday: Which Exponent?
Dusk settles over the Glass House, and Nova's light throws a soft circle on the duckweed trough.
"Five fronds on day zero, ten on day one, twenty on day two," Wren reads from the log. "Our doubling model from last week."
"The counting grid over the trough has 640 squares," Comet says. "When does the duckweed fill it? What can we make of that?"
Wren writes 5 × 2ᵗ = 640 on the chalkboard. "So 2ᵗ = 128. Which power of 2 is 128?"
Nova hovers closer. "Would you like a hint? Count the doublings on your fingers."
"2, 4, 8, 16, 32, 64, 128. Seven doublings," Comet says. "So seven days."
"That question has a name," Wren says. "Which exponent gives 128? That number is called a logarithm."
A logarithm is an exponent
The question "which power of 2 gives 128?" has the answer 7. We write log₂ 128 = 7.
Read it as "log base 2 of 128 equals 7." The small 2 is the base. The answer is always an exponent.
In general, log_b N = k means exactly the same thing as bk = N. The two sentences swap roles, nothing more.
Base 10 is so common that log 1,000 with no small number means log₁₀ 1,000, which is 3.
A solved problem to study
- The crew's model: 5 × 2ᵗ = 640, where t is days since day zero.
- Divide both sides by 5: 2ᵗ = 128.
- Ask the logarithm question: t = log₂ 128.
- Find the exponent: 2⁷ = 128, so t = 7.
- Check: 5 × 2⁷ = 5 × 128 = 640. True.
- Answer: the duckweed covers the grid on day 7.
Notice the order. First divide away the starting count, then ask the logarithm question. Dividing first keeps the equation simple.
Every number here is the crew's own trough count from the log, not a fact about real duckweed.
Powers of 2 and their logarithms
| Power | Value | Logarithm |
|---|---|---|
| 2⁰ | 1 | log₂ 1 = 0 |
| 2¹ | 2 | log₂ 2 = 1 |
| 2² | 4 | log₂ 4 = 2 |
| 2³ | 8 | log₂ 8 = 3 |
| 2⁴ | 16 | log₂ 16 = 4 |
| 2⁵ | 32 | log₂ 32 = 5 |
| 2⁶ | 64 | log₂ 64 = 6 |
| 2⁷ | 128 | log₂ 128 = 7 |
| 2⁸ | 256 | log₂ 256 = 8 |
| 2⁹ | 512 | log₂ 512 = 9 |
| 2¹⁰ | 1,024 | log₂ 1,024 = 10 |
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| log₂ 8 = 3 because 2³ = 8. | ? |
| 2 × 2 × 2 × 2 × 2 = 32 | ? |
| log₁₀ 100 = 10. | ? |
| A logarithm is an exponent. | ? |
| 5 × 2 × 2 × 2 = 80 | ? |
Cover the exponents. Point at a value and say its log base 2 out loud, then uncover to check.
Strong start. Tomorrow a slower trough gives an answer that is not a whole number, and the logarithm still finds it.