1. Isolate $e$: You must divide by $P$ before you can use the $\ln$ button.
2. Unlock the Exponent: $\ln(e^x) = x$. The $\ln$ "brings down" the power.
3. The Magic Number: $\ln(2) \approx 0.693$. Use this for doubling time.
4. Check for Reality: Time ($t$) should always be a positive number.
Solve for $x$ without a calculator where possible.
$\ln(e^7) = x$
$e^x = 15$
$3 \cdot e^x = 30$
Use the formula: $t = 0.693 / r$.
The Fast Growth: A bacteria colony grows continuously at a rate of 50% per hour ($r=0.5$). How long does it take for the population to double?
The Slow Fruit: A Kingdom Fund grows at 5% per year ($r=0.05$). How long until the gold doubles in size?
If you have $\$1$ and it grows to $\$1,000,000$, will the Natural Log ($\ln$) of the result be a large number like a million, or a small number? Why?
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(Hint: $\ln(1,000,000) \approx 13.8$. Does 13.8 represent the money or the time?)
The Harvest Target: You start with 100 seeds. They grow continuously at 15% per day ($r=0.15$). On which day will you have 1,000 seeds?
The Decay Target: A medicine (100mg) decays at a rate of 20% per hour ($r=-0.20$). How long until only 10mg is left in the body?
We know $\ln(2)$ is for doubling.
1. What is the "Magic Number" for **Tripling** ($A = 3P$)?
2. If a forest grows at 2% per year, how long will it take to triple in size?
Objective: Explain "Doubling Time" to a younger student.
The Activity: Use two cups.
Cup 1: Put 2 candies in.
Cup 2: Tell them this cup "doubles" every 10 minutes.
The Question: "If I want 16 candies, and I have 2 right now... how many 'doubles' (10-minute jumps) do I have to wait for?"
The Lesson: "In high school math, we call this a 'Natural Log.' It's the clock that tells us how long we have to wait for the candy to multiply."
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