First — talk for 2 minHas anyone in your family voted, registered to vote, or had trouble voting? Or thought about voting for the first time? If no one, Mr. Bennett will share.
Why does a billion collapse to a single digit?
To vote in any US election, your name has to be on a state's voter roll. States “clean” those rolls every cycle — they take off people who moved, died, or registered twice. That is a purge. Some purges are routine. Some take eligible voters off and never tell them.
Between 2020 and 2022, US states removed about 19,000,000 names from voter rolls. In one wave in September 2023, Ohio purged 154,995 names. Wisconsin tried to drop 234,000 in late 2019 (a court halted it). Georgia cut about 1,500,000 names from 2012–2016. Different states, very different sizes — but written out, all those zeros start to blur together.
Here is the move: a \(\log(x)\) of a count tells you how many digits the number has, almost exactly. \(\log(154{,}995)\) lands between 5 and 6. \(\log(19{,}000{,}000)\) lands between 7 and 8. One digit of difference on the log scale — how many times bigger is that, and what does that compression let you see that the raw count hides?
Worked Example #1 — Sandwich \(\log(154{,}995)\) (Ohio purge, Sep 2023)
ProblemOhio purged 154,995 names in one wave. Find two consecutive integers that sandwich \(\log(154{,}995)\). No calculator.
DataThe base is 10 (no base shown means base 10). Powers of 10: \(10^{3}=1{,}000\), \(10^{4}=10{,}000\), \(10^{5}=100{,}000\), \(10^{6}=1{,}000{,}000\), \(10^{7}=10{,}000{,}000\). Use \(\log_{b}(x) = n\) means \(b^{n} = x\).
Word Bank
base
exponent
sandwich
larger
smaller
argument
denominator
between
Number Bank
10
5
6
100,000
1,000,000
154,995
15
10,000
| # | Reasoning (fill in) | Math (fill in numbers) |
|---|
| 1 |
Note the . \(\log\) with no base means base 10. |
\(b = \) |
| 2 |
The number 154,995 is the of the log — the value going in. |
input \(= \) |
| 3 |
Find the largest power of 10 that is than 154,995. |
\(10^{5} = \) \(<\) 154,995 ✓ |
| 4 |
Find the smallest power of 10 that is than 154,995. |
\(10^{6} = \) \(>\) 154,995 ✓ |
| 5 |
The log lives those two exponents. |
\(\log(154{,}995)\) is between and |
Page 1 of 3 · Evaluating Logarithmic Expressions · Algebra 2 · Unit 5 · Lesson 11
Worked Example #2 — Compare a purge to an intervention
ProblemOregon's first year of Automatic Voter Registration added 272,000 new voters. Sandwich \(\log(272{,}000)\). Then say which is bigger on the log scale: Oregon's add (272,000) or Ohio's purge (154,995).
Data\(10^{5} = 100{,}000\), \(10^{6} = 1{,}000{,}000\). From WE#1, \(\log(154{,}995)\) is between 5 and 6 (calculator: about 5.19). When two logs sit in the same integer range, compare the inputs directly to break the tie.
Word Bank
power
exponent
larger
same
smaller
range
numerator
input
Number Bank
10
5
6
100,000
1,000,000
272,000
5.43
6.43
| # | Reasoning (fill in) | Math (fill in numbers) |
|---|
| 1 |
Note the base — same as WE#1. |
\(b = \) |
| 2 |
The largest of 10 still less than 272,000. |
\(10^{5} = \) |
| 3 |
The smallest power of 10 than 272,000. |
\(10^{6} = \) |
| 4 |
The log sits between those two s. |
\(\log(272{,}000)\) is between and |
| 5 |
Both logs land in the integer range (5 to 6). With the same base, the larger input gives the larger log. |
272,000 \(>\) 154,995, so \(\log(272{,}000) \approx \) |
Real-Data Work Time — fill in from your sandwich work
Q1 (Wisconsin, 234,000): \(\log(234{,}000)\) is between and , because \(10\) \(<\) 234,000 \(<\) \(10\).
Q2 (Georgia, 1,500,000): \(\log(1{,}500{,}000)\) is between and .
Q3 (US national, 19,000,000): \(\log(19{,}000{,}000)\) is between and .
Q4 (calculator check): Type \(\log(154{,}995) \approx \) . Type \(\log(19{,}000{,}000) \approx \) . Difference of about on the log scale = about times bigger in raw count.
Conclude
Which intervention would you push for — AVR, Same-Day, or Pre-Reg at 16/17?
The math just compressed four very different state purges into the same one-digit window. Now use the same trick on the three real interventions. Compare on the log scale, then defend a choice for Pennsylvania (which has none of the three as of 2026).
Oregon AVR — given (year 1)
New voters: 272,000
\(\log(272{,}000)\) sits between and
Same order as Ohio's purge? (yes / no)
Pre-Reg at 16/17 — compute
Effect: +9 percentage points youth turnout.
If 100,000 PA youth would otherwise vote, +9 pp adds about voters.
\(\log\) of that addition sits between and
Reflect: A single AVR rollout adds about \(10^{5}\) voters in year 1 — same order as a single purge wave. Which intervention would you push for in PA, and which number convinced you?
Page 2 of 3 · Evaluating Logarithmic Expressions · Algebra 2 · Unit 5 · Lesson 11
Tier 2 · General Academic Words
- evaluate — find the value of something by using the rules.
- expression — a piece of math made of numbers, letters, and operations (no equals sign needed).
- value — the number an expression works out to.
- compress — squeeze something big into something smaller (logs compress millions into one digit).
- order of magnitude — the size class of a number, by how many digits it has.
- estimate — a careful guess based on what you already know.
- verify — check that an answer is correct by trying it a second way.
- roll — an official list of names (a voter roll lists everyone allowed to vote in a place).
- purge — to take names off a list, all at once.
- wave — a single round of something (one purge wave means one big batch removed).
- intervention — an action taken to change what would have happened.
- cycle — one full round of something that repeats (an election cycle is two years).
Tier 3 · Logarithm & Civic Words
- logarithm — the exponent you have to raise a base to in order to get a given number. \(\log_{2}(8) = 3\) because \(2^{3} = 8\).
- base — the number being raised to a power (in \(\log_{2}\), the base is 2).
- argument — the number going into the log (in \(\log(154{,}995)\), the argument is 154,995).
- exponent — the small number a base is raised to. A logarithm is an exponent.
- log base 10 — \(\log_{10}(x)\), spoken “log base 10 of \(x\)”: the exponent on 10 that gives \(x\).
- common log — the log with base 10. Written \(\log(x)\) with no base shown.
- sandwich — trap a value between two integer powers of the base (\(10^{5} < 154{,}995 < 10^{6}\)).
- voter roll — the official list of registered voters in a state or county.
- Automatic Voter Registration (AVR) — a state law that signs eligible people up to vote when they get a driver's license, unless they opt out.
- Same-Day Registration (SDR) — you can register and vote on the same day at the polls.
- Pre-Registration — a state law letting 16- and 17-year-olds sign up early so they're auto-active at 18.
- turnout — the share of registered voters who actually vote in an election.
Page 3 of 3 · Evaluating Logarithmic Expressions · Algebra 2 · Unit 5 · Lesson 11