First — talk for 2 minHas anyone ever felt an earthquake, even a small one? Or tested soil for a garden, or worried about water at home? If no one, Mr. Bennett will share.
Two log scales hiding in the dirt
A farmer in Lancaster County, PA sends a soil sample to the lab. The report comes back with a single tiny number: hydrogen ion concentration of \(10^{-5}\) moles per liter. From that one number, the lab can tell her whether her field will grow chestnut trees, blueberries, or alfalfa — or whether she needs to spread lime first. Pick the wrong crop and the harvest fails. The lab’s tool for shrinking that strange-looking concentration into a friendly number on a 0–14 scale is a logarithm.
Two hundred miles west, in Youngstown, Ohio, the ground had not produced a single recorded earthquake since 1776. Then in late 2010 a deep injection well opened, and trucks from Pennsylvania began filling it with fracking wastewater. Thirteen days later the ground started moving. Over the next year, instruments recorded 109 earthquakes. The biggest, on December 31, 2011, registered magnitude 4.0 on the Richter scale. Same kind of math — a logarithm — turning a tiny seismograph wiggle into a single, friendly number.
Two stories. Two states. Two log scales. Why does the same hidden math run both — and how much bigger is “just one step” on a log scale than it sounds?
Worked Example #1 — pH from hydrogen ion concentration
ProblemThe Lancaster farmer’s soil sample has hydrogen ion concentration \(c = 10^{-5}\) mol/L. Find the pH using \(\text{pH} = -\log_{10}(c)\).
DataPure water has \(c = 10^{-7}\) and pH 7. Smaller pH = more acidic. Chestnut wants pH 5.5–6.5; alfalfa wants pH 6.5–7.5. The pH is the exponent of 10 in c, with the sign flipped.
Word Bank
substitute
exponent
flip the sign
power of 10
logarithm
multiply
acidic
read off
Number Bank
10
5
−5
7
−7
100
1/100
3.16
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Write the concentration as a clean . The lab already gave it that way. |
\(c = 10^{\,\square}\), with \(\square = \) |
| 2 |
the exponent. By the rule \(\log_{10}(10^{x}) = x\), the log just hands it back. |
\(\log_{10}(10^{-5}) = \) |
| 3 |
by the coefficient \(-1\) sitting in front of the log. The negative coefficient is what turns the negative exponent into a positive pH. |
\(\text{pH} = -(\,\square\,)\), with \(\square = \) |
| 4 |
Compute. Two negatives cancel, so the pH is positive. |
\(\text{pH} = \) |
| 5 |
Compare to pure water at pH 7. The farmer’s soil is two pH steps lower, so it is times more than water. |
\(\dfrac{10^{-5}}{10^{-7}} = 10^{2} = \) |
Page 1 of 3 · Logarithms in the Ground · Algebra 2 · Unit 5 · Lesson 18
Worked Example #2 — Richter magnitude from seismograph displacement
ProblemThe Youngstown 2011 induced quake moved a seismograph needle by displacement \(d = 10^{-3}\) m at 100 km from the epicenter. Find its Richter magnitude using \(M = \log_{10}(d) + 7\).
DataThe reference distance is 100 km. Each whole step on \(M\) means 10× the displacement. \(d\) is in meters; the +7 shifts the scale so a 1-micrometer microquake (\(10^{-6}\) m) reads as \(M = 1\).
Word Bank
substitute
exponent
add
power of 10
logarithm
displacement
magnitude
read off
Number Bank
10
7
−3
3
4
100
2.097
9.1
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Write the displacement as a clean . The instrument already reports it that way. |
\(d = 10^{\,\square}\) m, with \(\square = \) |
| 2 |
Read off the using the rule \(\log_{10}(10^{x}) = x\). |
\(\log_{10}(10^{-3}) = \) |
| 3 |
7 to that result. The +7 is the shift that turns negative-exponent meters into positive Richter numbers. |
\(M = -3 + \) |
| 4 |
Compute. This matches the M 4.0 reported on Dec 31, 2011 in Youngstown. |
\(M = \) |
| 5 |
Check the scale: the 2004 Indonesia quake had d ≈ 125 m. Use a calculator: \(\log_{10}(125) \approx \) , then add 7. |
\(M \approx 2.097 + 7 = \) |
Activity · Bend the Rate (work with your partner)
Pick a story. Name the rate. Defend the wrench.
Story 1 — Soil: The farmer’s field is pH . To plant alfalfa (target pH 6.5) she must raise pH by units, which means cutting H+ by a factor of \(10^{1.5} \approx \) .
Story 2 — Quakes: The Youngstown event was M . A typical PA fracking microquake is M 2. The ground moved times more in Youngstown, because \(\dfrac{10^{4-7}}{10^{2-7}} = 10^{\,\square}\) with \(\square = \) .
Pick 2 interventions from the menu for your story and explain whether your bundle moves the rate enough.
○ USDA EQIP cost-share — covers 50–75% of liming cost (NRCS CP-590)
○ OH/PA “stoplight” rule — halt injection at M ≥ 2 within 5 km; cut induced quakes ~80% (Schultz 2020)
○ Subsidized soil testing — +15–25% better lime targeting (Penn State AASL)
○ PA Act 13 traffic-light protocol — auto-shutdown at M 2 in 5 km (2017)
My defense:
Conclude · The Wrench
Same math, two stories. Whose choice sets the rate?
A 1-unit step on a log scale = a 10× step in the world. Apple juice (pH 4) has 1,000× the H+ of pure water (pH 7). A magnitude-7 quake shakes the ground 10× harder than a magnitude-6, and 100× harder than a magnitude-5. The math doesn’t care which story it’s in — but the rate is set by people.
Soil pH ladder — fill in
Pure water: pH 7, [H+] = \(10^{-7}\)
Coffee (pH 5): [H+] = mol/L
Apple juice (pH 4): [H+] = mol/L
Apple juice has × the H+ of water.
Richter ladder — fill in
M 2 (PA fracking microquake): \(d = \) m
M 4 (Youngstown 2011): \(d = 10^{-3}\) m
M 7 (Haiti 2010): \(d = \) m
Haiti shook × harder than Youngstown.
Reflect: Carbon-14’s half-life is set by physics. But the pH of a Lancaster field can be raised with lime, and the magnitude of a Youngstown quake depends on whether the well next door is open. Whose choice sets each rate — and where would your wrench go?
Page 2 of 3 · Logarithms in the Ground · Algebra 2 · Unit 5 · Lesson 18
Tier 2 · General Academic Words
- estimate — a careful guess based on what you know, often using a formula.
- sample — a small piece taken from a bigger thing so it can be tested.
- ratio — one number divided by another, used to compare them.
- scale — a way of measuring something on a number line.
- compress — squeeze a wide range of numbers into a smaller, friendlier range.
- shift — slide a value up or down by adding or subtracting a constant.
- concentration — how much of something is packed into a given amount of liquid.
- hidden — not easy to see at first, but really there if you look.
- magnitude — the size or strength of something, often as a single number.
- reference — a fixed point you measure other things from (here: 100 km).
- policy — a rule or plan made by a government, company, or group.
- intervention — an action taken to change how something is going.
- induced — caused by something humans did, not by nature on its own.
- permit — an official paper that allows something to happen.
Tier 3 · Soil, Quake, & Math Words
- logarithm — the answer to the question “what exponent do I put on the base to get this number?” If \(b^{x} = y\), then \(\log_{b}(y) = x\).
- log base 10 — \(\log_{10}\). Asks “what power of 10 gives this number?” Used in pH and Richter.
- log base \(b\) — \(\log_{b}\). Asks “what power of \(b\) gives this number?” Example from the warm-up: \(\log_{4}(\tfrac{1}{16}) = -2\) because \(\tfrac{1}{16} = 4^{-2}\).
- inverse — an operation that undoes another. The log undoes the exponent: \(\log_{10}(10^{x}) = x\).
- exponent — the small raised number that says how many times to multiply the base by itself.
- pH — a 0–14 scale for acidity. Lower means more acidic; 7 is neutral; higher means more basic.
- hydrogen ion (H+) — the tiny charged particle whose concentration sets a liquid’s acidity.
- acidic — having more H+ ions than pure water; pH less than 7.
- basic — having fewer H+ ions than pure water; pH greater than 7.
- lime (agricultural lime) — ground-up limestone (calcium carbonate) spread on fields to raise pH.
- Richter scale — a log-base-10 scale for earthquake size, built from the seismograph displacement at 100 km.
- seismograph — an instrument that measures how much the ground moves during a quake.
- displacement — how far the seismograph needle moved (in meters).
- epicenter — the point on the ground directly above where an earthquake started; Richter readings use a 100 km reference distance from it.
- induced seismicity — earthquakes caused by human activity, like injecting wastewater deep underground.
- injection well — a deep well used to dispose of fracking wastewater by pumping it underground.
- fracking — pumping pressurized fluid into rock to crack it open and release oil or gas.
- traffic-light protocol — a rule that auto-halts injection if a quake of a certain magnitude occurs nearby.
Page 3 of 3 · Logarithms in the Ground · Algebra 2 · Unit 5 · Lesson 18