Understand the definition of the Natural Logarithm ($ln$). It is a logarithm with base $e$. If $e^x = y$, then $\ln(y) = x$. Prepare to teach that $ln$ is the "Time machine" for the PERT formula. In the Kingdom, everything has a "Doubling Time" and a "Harvest Date." $ln$ is how we find them.
In Lesson 17.2, we discovered the number **$e$**—the "Abundance Constant." We learned that life grows continuously, according to the formula $A = Pe^{rt}$. But in that lesson, we always knew the "Time" ($t$). We were looking forward into the harvest.
But there are times when we know the harvest we want, and we ask, "Lord, how long?" How long until the leaven fills the lump? How long until the seed becomes a tree? How long must I wait for the breakthrough?
To answer these questions, we need the Natural Logarithm ($ln$). If $e$ is the "Breath of Life," then $ln$ is the "Unveiling of the Breath." It is the key that unlocks the exponent of time.
The Psalmist prayed, "Teach us to number our days, that we may gain a heart of wisdom" (Psalm 90:12). $ln$ is the mathematical way we "number our days." it tells us the relationship between the growth we see and the time that has passed. It is the math of Patience and Fulfillment.
Today, we learn to "Undo" the power of $e$. we will see that every continuous process has a rhythm, and the Natural Log is the beat of that rhythm.
The Rupture: The student tries to calculate $\ln(-10)$.
The Repair: "Counselor, you are trying to find the 'Time' of a non-existent life! $e^x$ can never be negative. No matter how much time passes, you cannot grow into 'less than nothing.' Therefore, the Natural Log is blind to the negative. It only sees what exists in the light of the positive. Clear your mind of the 'void' and look only at the 'being'."
1. $A = Pe^{rt}$ (The Equation)
2. $A/P = e^{rt}$ (Isolate the $e$)
3. $\ln(A/P) = rt$ (Take the Natural Log)
4. $t = \ln(A/P) / r$ (Find the Time)
1. **Isolate First**: Never take the $ln$ until the $e$ is by itself.
2. **Check the Log**: $\ln(A/P)$ will be positive if you are growing ($A > P$) and negative if you are decaying ($A < P$).
3. **Consistency**: Time ($t$) should always be positive. If you get a negative time, you may have switched $A$ and $P$.
The older student should use the "Plant Growth" metaphor. "If a sunflower grows a little bit every second, and it needs to be 6 feet tall to touch the sky... how do we know when it will get there?"
The older student must explain: "In my math, I have a 'Time-Key' called $ln$. It looks at how tall the flower is now and how fast it's growing, and it tells me exactly what day I should bring my camera to take a picture of the top."
"This is why we don't worry about the future. We know the rhythm of God's growth."
A seed of the Word is planted in a heart. It grows **continuously** at a rate of 20% per day ($r=0.20$).
Task: How many days will it take for the seed to reach "100-Fold" (100 times its original size)? Show the steps using $ln$.
Theological Requirement: Notice that the answer is not "100 days." Multiplication is faster than addition. Reflect on why God's "suddenlies" in the Bible often happen after a season of quiet, continuous growth. How does the Natural Log help us "wait with wisdom"?
The number $\ln(2) \approx 0.693$ is the most important number in growth. If you multiply your growth rate ($r$) by 100, and divide 70 by that number, you get the **Doubling Time**.
Example: 7% growth. $70 / 7 = 10$ years. It will double in about 10 years.
This is a "Prophetic Shortcut." It allows you to see the future of any system in an instant. It is the math of the "Quickening."
The Natural Log ($ln$) is often intimidating because it doesn't look like a number. Remind the student: **$ln$ is just a Log with a base that is alive ($e$).**
Ensure they understand that $\ln(y)$ is the Power. If $\ln(y) = 5$, it means $e^5 = y$. The "answer" of a log is always the "exponent" of the power. This connection is the anchor of the entire edition.