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Algebra 2 9-12 / Week 07 / Tuesday
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Week 07 · Logarithms

Tuesday

Solving with logarithms
// The exponent, found backwards
⏱ about 20 min

Tuesday: Solving With Logarithms

"The second trough is slower," Comet says, reading the log. "Twelve fronds, times 1.5 each day. Its grid has 300 squares."

Wren writes 12 × 1.5ᵗ = 300 and divides. "1.5ᵗ = 25. Which power of 1.5 is 25? Not a whole number."

"Then we guess," Comet says. "1.5⁷ is about 17. 1.5⁸ is about 25.6. Between seven and eight days."

"Close, but what does the evidence say exactly?" Wren asks.

Nova projects a glowing key. "Would you like a hint? Your calculator has a log key for this."

"log 25 divided by log 1.5," Wren says, tapping it in. "About 7.9. So the grid fills partway through day eight."

"Two ways," Comet says. "A table gets close, and the logarithm lands it."

Way one: a table

Make a table of t and 12 × 1.5ᵗ. Go down the rows until the value passes 300. The answer sits between two rows.

A table is honest and slow. It finds the whole days on either side, never the exact moment.

t (days)12 × 1.5ᵗPast 300?
591.1no
6136.7no
7205no
8307.5yes
9461.3yes

Way two: a logarithm

  1. Start with 12 × 1.5ᵗ = 300.
  2. Divide both sides by 12: 1.5ᵗ = 25.
  3. Write the logarithm question: t = log base 1.5 of 25.
  4. A calculator has a log key for base 10. Any base works by dividing: t = log 25 ÷ log 1.5.
  5. Tap it in and round: t ≈ 7.9 days.
  6. Check: 12 × 1.5 to the 7.9 is close to 300. The table agrees, between 7 and 8.

Why does dividing two logs work? Taking log of both sides of 1.5ᵗ = 25 gives t × log 1.5 = log 25.

That uses one rule: the log of a power is the exponent times the log. Divide by log 1.5 and t is alone.

The crew keeps the table in the log as a check. If the two ways disagree, something was typed wrong.

The general recipe

Any equation a × bᵗ = d is solved the same way. Divide by a, take the log of both sides, divide by log b.

Round only at the end, and say how many places you kept. The helper answers here keep one place.

SOLVE WITH A LOGARITHM
  • Read the question.
  • Tap your answer.
The second trough: 12 fronds, times 1.5 each day, a 300-square grid. After how many days is the grid covered? (Round to 1 place.)
The first trough: 5 fronds doubling each day, a 640-square grid. After how many days is the grid covered?
A seed tray count starts at 20 and triples each day in the crew's model. After how many days does it reach 540?
A moss patch count starts at 8 and multiplies by 1.2 each day in the crew's model. After how many days does it reach 40? (Round to 1 place.)
SOLVE 12 × 1.5ᵗ = 300, IN ORDER
  • ?Divide both sides by 12: 1.5ᵗ = 25
  • ?Take the log of both sides: t × log 1.5 = log 25
  • ?Divide by log 1.5: t = log 25 ÷ log 1.5
  • ?Tap it in and round: t ≈ 7.9
  • ?Check against the table: between day 7 and day 8
WHY THIS EXERCISEThe same five moves solve every a × bᵗ = d. Doing them in order keeps the algebra clean.
StatementTrue or false?
To solve 12 × 1.5ᵗ = 300, first divide both sides by 12.?
log 25 ÷ log 1.5 is the same as log 25 minus log 1.5.?
12 × 25 = 300?
A table of values can find the exact day the grid fills.?
The log of a power equals the exponent times the log of the base.?
WHY THIS EXERCISEThe log rules are exact. The table is the check that catches a typing slip.
Wren's table shows 12 × 1.5ᵗ passing 300 between two whole days. The logarithm gives about 7.9. Which whole day comes first, before the grid fills? Type the number.
WHY THIS EXERCISEThe table and the logarithm must agree. Reading both is how the crew trusts a decimal answer.
Try it
Pick a start count and a daily factor of your own. Choose a target and solve two ways: table, then logarithm.
Write both answers side by side in your log. Circle the whole days that bracket the decimal.

Excellent. Tomorrow the thermometer comes out and warm water cools in the Glass House Lab.

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