1. Warm-Up Algebra 2 Honors 2/22/18
Simplify
Ln e40x Ln e(x+4) ln e 2x + ln ex
4. Try solving:
ey=ex+3
3 minutes to do
2 minutes to go over
5 total

2. The half-life of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of decay. Natural decay is modeled by the function below.
2 minutes
7 total

3. Plutonium-239 (Pu-239) has a half-life of 24,110 years
Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?
Step 1 Find the decay constant for plutonium-239.
N(t) = N0e–kt
Use the natural decay function.
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𝑁 = N0e–k(24,110)
Substitute 24,110 for t and 𝑁 for N(t) because half of the initial quantity will remain.

4. ln 1 2 =ln 𝑒 −24110𝑘 ln 2 −1 = -24110k − ln 2 = -24110k
Simplify and take ln of both sides
ln 2 −1 = k
Write ½ as 2-1, and simplify the right side
− ln 2 = k
ln2–1 = –1ln 2 = –ln 2
5 minutes
17 minutes
k= 𝑙𝑛
Solve for k k≈

5. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?
Step 2 Write the decay function and solve for t.
How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?
N(t) = N0e– t
Substitute for k.
Substitute 1 for N0 and 0.01 for N(t).
0.1 = 1e– t
Take ln of both sides.
ln 0.1 = ln e– t
Simplify.
ln 0.1 = – t
5 minutes
22 total
𝑡= 𝑙𝑛0.1 − ≈80,000
It takes approximately 80,000 years to decay.

6. Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg.
Step 1 Find the decay constant for Chromium-51.
N(t) = N0e–kt
Use the natural decay function.
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25 minutes
= 650e–k(28)
Substitute 24,110 for t and 𝑁 for N(t) because half of the initial quantity will remain.

7. ln 1 2 =ln 𝑒 −28𝑘 ln 2 −1 = -28k − ln 2 = -28k ln2–1 = –1ln 2 = –ln 2
Simplify and take ln of both sides
ln 2 −1 = -28k
Write ½ as 2-1, and simplify the right side
− ln 2 = -28k
ln2–1 = –1ln 2 = –ln 2
5 minutes
30 total
k= 𝑙𝑛2 28
Solve for k k≈

8. How long does it take for 650 mg of chromium-51 to decay to 200 mg.
Step 2 Write the decay function and solve for t.
How long does it take for 650 mg of chromium-51 to decay to 200 mg.
N(t) = N0e– t
Substitute for k.
Substitute 650 for N0 and 200 for N(t).
200= 650e– t
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= e– t
Divide by 650 on both sides

9. 𝑙𝑛 = ln 𝑒 − 𝑡
Take ln of both sides.
ln = – t
Simplify.
5 minutes
40 minutes
𝑡= 𝑙𝑛 − ≈47
It takes approximately 47 days to decay.

10. Solutions with certain pH’s are classified by: less than 7 are acidic
Chemists define the acidity or alkalinity of a substance according to the formula 
"pH = –log[H+]" 
where [H+] is the hydrogen ion concentration, measured in moles per liter.
Solutions with certain pH’s are classified by:
less than 7 are acidic
greater than 7 are basic
a pH of 7 (such as pure water) are neutral.
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11. 2 minutes
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12. This is less than 7 so it’s acidic
Suppose that you test apple juice and find that the hydrogen ion concentration is [H+] = Find the pH value and determine whether the juice is basic or acidic. pH = –log[H+] pH = –log(.0003) =
This is less than 7 so it’s acidic
3 minutes
48 minutes

13. This is greater than 7 so it’s basic
You test some ammonia and determine the hydrogen ion concentration to be [H+] = 1.3 × 10–9. Find the pH value and determine whether the ammonia is basic or acidic. pH = –log[H+] pH = –log(1.3 × 10–9) = …
This is greater than 7 so it’s basic
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50 minutes