Algebra 2 · U5 L17 · Logarithmic Functions

First — talk for 2 minHave you ever planted a tree or watched one grow? How long until anything visible happened? Has anyone in your family worked land or grown food? If no one, Mr. Bennett will share.

The shape of a slow ceiling

Plant a chestnut tree in South Jersey today, and you wait five years for the first nut. From there, in a clean model, the harvest doubles every year — a fast bear-up phase that races toward a mature ceiling near 2,000 pounds per acre. After that, no matter how patient the grower is, the trees can't push past their own biology.

Below the canopy, something slower is happening. With no-till farming, cover crops, manure, and tree litter, soil organic carbon climbs — but each step takes a tenfold increase in compounded effort. Eventually the soil's mineralogy caps how much carbon it can hold. Two real things, two different bases, the same shape: shoot up fast, then bend hard toward flat.

A 16-year-old grower can't change the base of either curve. But cost-share programs (EQIP), conservation rentals (CRP), and a member-owned co-op (KTCC) all raise the ceiling itself, or fund the slow climb until mature yield arrives. What kind of function shoots up fast and then nearly flattens — and how much higher can people actually push the ceiling?

Worked Example #1 — Read $t = \log_{2}(P)$ (chestnut yield, forward)

ProblemThe chestnut model uses $f(x) = \log_b(x)$ with base 2. Let t = years of doubling and P = yield in thousand lbs/acre. Find t when P = 50 using $t = \log_{2}(P)$.

DataThe orchard hits 1 thousand lbs/acre at year 0, then doubles each year. The function's $P > 0$ rule means we cannot take the log of zero or a negative yield. Use $\log_b(x) = y \iff b^y = x$: the log is the exponent.

Word Bank exponent substitute base double multiply between divide calculator

Number Bank 5 6 2 50 32 64 5.64 25

#	Reasoning (fill in)	Math
1	Read the equation. The output t is the that b = 2 must be raised to so the result equals P.	$t = \log_{2}(P)$, with $b = $
2	$P = 50$ into the equation.	$t = \log_{2}(\,\square\,)$, with $\square = $
3	Find the two powers of 2 that bracket 50. $2^{5} = $ and $2^{6} = $ . So t lies 5 and 6.	$2^{5} = 32 < 50 < 64 = 2^{6}$
4	Use a : $\log_{2}(50) = \log(50) / \log(2)$. Each time P doubles, t increases by 1.	$\log_{2}(50) \approx $
5	State the answer in context: about years of doubling carry the orchard from 1 to 50 thousand lbs/acre.	$t \approx 5.64$ years

Page 1 of 3 · Logarithmic Functions · Algebra 2 · Unit 5 · Lesson 17

Worked Example #2 — Invert $\log_{10}$ (soil carbon, backward)

ProblemBelow the orchard, soil carbon follows $\log_{10}(x)$. Define $f(p) = \log_{10}(p)$, where p = soil carbon stock relative to baseline 1. If $f(p) = 5$, find p. Then say what 5 "tenfoldings" actually means.

DataEach unit of f means the input p has been multiplied by 10. The natural log $\ln(x)$ uses base e; here we use base 10 (the common log). The domain rule $p > 0$ still applies — soil stock cannot be negative.

Word Bank exponent tenfold substitute rewrite domain multiply add saturation

Number Bank 5 10 100,000 50,000 100 1.7 10,000 1

#	Reasoning (fill in)	Math
1	$f(p) = 5$ into the equation $f(p) = \log_{10}(p)$.	$\log_{10}(p) = $
2	in exponential form using $\log_b(x) = y \iff b^{y} = x$. The base is 10; the exponent is 5.	$p = \,\square^{\,\square}$, $\square = $ , $\square = $
3	Compute $10^{5}$. Each of effort multiplies the soil stock by 10, five times in a row.	$p = 10^{5} = $
4	Check the : $p > 0$ (positive). 100,000 satisfies the rule, so the answer is valid.	$p = 100{,}000 > 0$ ✓
5	Read the meaning: the input had to climb 100,000-fold for f to add up to 5. This is the shape — tenfolds get harder.	$p = 100{,}000$ (a 100,000-fold climb of soil carbon)

Activity · What Raises the Ceiling? (work with your partner)

Pick a lever. Name the number. Defend the move.

Real ceiling (chestnut): mature yield ≈ lbs/acre, reached around year 15. Real ceiling (soil carbon): set by soil mineralogy, not effort.

Years of low yield: the grower needs cash from year 5 to year while the curve climbs. Closing this gap is the wrench.

Pick 2 levers from the menu: circle two and explain whether your bundle pays for the climb and raises the ceiling.

○ EQIP Practice 612 (tree/shrub) — covers 45–75% of NJ planting cost (NRCS Mid-Atlantic, FY 2025)

○ EQIP Practice 311 (alley cropping) — $20.80–$88.50/acre/yr (NRCS state schedules, FY23)

○ CRP rental — ~$50–$300/acre/yr for 10–15 years (USDA FSA)

○ Carbon credits (regen-ag) — ~$55–$110/acre/yr at $15–$30 / tCO₂e (voluntary market, 2024–25)

○ KTCC co-op aggregator — raises farm-gate $/lb on chestnut flour, hickory + hazel oil (ktcc.coop)

○ Climate-Smart Commodities — layered per-acre stewardship payments (USDA, 2022–2027)

My defense:

Conclude · The Wrench

The base sets the shape. What sets the ceiling?

Both $t = \log_{2}(P)$ and $f(p) = \log_{10}(p)$ shoot up early, then flatten. The flatness is biology and mineralogy — real ceilings, not bad effort. Below, name one biological ceiling and one policy lever. Then write your bundle's predicted revenue boost over the year-5-to-15 climb.

One real ceiling (biology or soil)
One policy lever that funds the climb

If your bundle held — close the gap Lever 1 ($/acre/yr):
Lever 2 ($/acre/yr):
Total over 10 years (year 5 → 15): $ /acre

Reflect: The shape of a log curve is set by the base. The height of the ceiling is set by people — programs, co-ops, and credits. Which lever would you push first if it were your 10-acre orchard, and why?

Page 2 of 3 · Logarithmic Functions · Algebra 2 · Unit 5 · Lesson 17

Tier 2 · General Academic Words

function — a rule that takes an input and gives back exactly one output.
inverse — the rule that undoes another rule. If one says “double it,” the inverse says “halve it.”
invert — to swap input and output of a rule (find the inverse).
domain — the set of all inputs a function is allowed to take.
ceiling — the top limit a value gets close to but does not pass.
baseline — the starting amount used as the comparison point.
saturation — the point where adding more effort barely changes the result.
compounded — multiplied again and again, with each step feeding the next.
tenfold — ten times as much; multiplied by 10.
intervention — an action taken to change how something is going.
lever — one specific thing you can push on to change the outcome.
stewardship — taking care of land, soil, or trees so they stay healthy for a long time.

Tier 3 · Logarithm, Orchard & Math Words

logarithmic function — a function of the form $f(x) = \log_b(x)$ that asks “what exponent on b gives x?” It is the inverse of an exponential function.
exponential function — a function of the form $g(x) = b^{x}$ where x is the exponent. The inverse of a log function.
exponential form — the form $b^{y} = x$, rewritten from $\log_b(x) = y$.
base (b) — the number being raised to a power. In $\log_{2}(x)$, the base is 2.
common logarithm — another name for $\log_{10}(x)$, the log with base 10. Often just written $\log(x)$.
log base 10 — the same as the common log; counts how many tenfoldings.
natural logarithm — the log with base e (about 2.718...). Written $\ln(x)$.
ln — short for natural log; $\ln(x) = \log_{e}(x)$.
chestnut orchard — a planted field of chestnut trees. Begins yielding around year 5; matures around year 15.
soil organic carbon (SOC) — the carbon stored in the dirt itself, in roots, leaves, and microbes.
EQIP — a USDA program (Environmental Quality Incentives Program) that pays farmers part of the cost of conservation practices.
CRP — a USDA program (Conservation Reserve Program) that rents farmland for 10–15 years to keep it in conservation use.
KTCC — Keystone Tree Crops Cooperative, a member-owned business that aggregates supply from many small chestnut growers so each grower gets a better farm-gate price.

Page 3 of 3 · Logarithmic Functions · Algebra 2 · Unit 5 · Lesson 17

#	Reasoning (fill in)	Math
1	Read the equation. The output t is the that b = 2 must be raised to so the result equals P.	\(t = \log_{2}(P)\), with \(b = \)
2	\(P = 50\) into the equation.	\(t = \log_{2}(\,\square\,)\), with \(\square = \)
3	Find the two powers of 2 that bracket 50. \(2^{5} = \) and \(2^{6} = \) . So t lies 5 and 6.	\(2^{5} = 32 < 50 < 64 = 2^{6}\)
4	Use a : \(\log_{2}(50) = \log(50) / \log(2)\). Each time P doubles, t increases by 1.	\(\log_{2}(50) \approx \)
5	State the answer in context: about years of doubling carry the orchard from 1 to 50 thousand lbs/acre.	\(t \approx 5.64\) years

#	Reasoning (fill in)	Math
1	\(f(p) = 5\) into the equation \(f(p) = \log_{10}(p)\).	\(\log_{10}(p) = \)
2	in exponential form using \(\log_b(x) = y \iff b^{y} = x\). The base is 10; the exponent is 5.	\(p = \,\square^{\,\square}\), \(\square = \) , \(\square = \)
3	Compute \(10^{5}\). Each of effort multiplies the soil stock by 10, five times in a row.	\(p = 10^{5} = \)
4	Check the : \(p > 0\) (positive). 100,000 satisfies the rule, so the answer is valid.	\(p = 100{,}000 > 0\) ✓
5	Read the meaning: the input had to climb 100,000-fold for f to add up to 5. This is the shape — tenfolds get harder.	\(p = 100{,}000\) (a 100,000-fold climb of soil carbon)