Friday morning the chalkboard wall is covered in log sentences from the week.
"Day 7 for the first trough. About 7.9 for the second," Comet reads. "3.1 minutes for the jug to halve its gap."
"Three different questions," Wren says. "What do you notice they share?"
"Each one asked for an exponent," Comet says. "Days, or minutes, hiding up in the power."
Nova sorts the sentences into two columns. "Would you like a hint? One column is powers, the other is logarithms."
"The same facts, written two ways," Wren says. "Today we read both columns without slowing down."
"And then I want to try one with base 10," Comet says. "A tally that multiplies by ten."
| Question | Equation | Logarithm | Answer |
|---|---|---|---|
| Days until the first trough is covered | 5 × 2ᵗ = 640 | t = log₂ 128 | 7 days |
| Days until the second trough is covered | 12 × 1.5ᵗ = 300 | t = log 25 ÷ log 1.5 | about 7.9 days |
| Minutes until the jug gap halves | 24 × 0.8ᵗ = 12 | t = log 0.5 ÷ log 0.8 | about 3.1 minutes |
| The inverse of f(x) = 2ˣ at 32 | 2ᵗ = 32 | t = log₂ 32 | 5 |
Every row has the same shape: a start, a factor, a target, and an exponent to find.
When the factor is 2 or 10 and the target is a whole power, count. Otherwise divide two logs on a calculator.
Nova's tally grid counts in tens: 1, 10, 100, 1,000. log₁₀ 1,000 = 3 because 10³ = 1,000.
If a count starts at 100 and multiplies by 10 each step, it reaches 100,000 after log₁₀ 1,000 = 3 steps.
Base 10 logarithms count the zeros on a power of ten. That is why the calculator key is base 10.
Solving 5 × 2ᵗ = 640 gives t = 7. Solving 5 × 2ᵗ = 2.5 gives t = -1, one day before day zero.
A negative time can be true algebra and still fall outside the log the crew kept. Say so when it happens.
| Statement | True or false? |
|---|---|
| log₁₀ 1,000 = 3. | ? |
| 10 × 10 × 10 = 1,000 | ? |
| log₂ 0.25 = -2 because 2⁻² = 0.25. | ? |
| To solve a × bᵗ = d you take the log first and divide by a last. | ? |
| The graph of y = log₁₀ x passes through (10, 1). | ? |
A full week of exponents found backwards. Tomorrow is Garden Day, and the family doubles beans on a tray.