First — talk for 2 minWhat's a job you've had, or one a family member has? Roughly what's the hourly pay? Take a wild guess at what the CEO of that company makes — we'll check it together.
How long would it take to catch up?
A Walmart cashier earns about $29,469 a year. The Walmart CEO, Doug McMillon, earns about $27.4 million. That is roughly 930 times the cashier's pay (AFL-CIO Executive Paywatch, 2024 fiscal year).
Imagine a cashier who gets a 3% raise every single year — the average raise across U.S. workers, according to the Bureau of Labor Statistics. Each year, their pay is multiplied by 1.03. After 50 years of working, their pay has grown by a factor of 1.03 to the 50th power, which is about 4.38.
So 50 years of 3% raises turns a $29,000 salary into about $129,000. That is a lot more than $29,000 — but it is nowhere near $27 million. The cashier's pay would have to grow by a factor of 930, not 4.38, to catch up. What rate would actually close the gap — and is it a rate any normal job has ever paid?
Worked Example #1 — Set up the catch-up equation
ProblemLet b = 1 + r be the yearly pay factor for an annual raise r. Write an equation that says "after 50 years of compounding by b, the cashier's pay has caught up to the CEO." Then estimate b.
DataCashier 2024 pay: $29,469. CEO 2024 pay: $27,400,000. Ratio: about 930. The reference table on Page 2 shows that \(b^{t} = k\) for several values of b at t = 50: at 14% it gives 700.23; at 15% it gives 1083.66.
Word Bank
multiply
factor
exponent
between
add
power
closer to
subtract
Number Bank
930
50
14%
15%
14.6%
700.23
1083.66
18.6%
| # | Reasoning (fill in) | Math |
|---|
| 1 |
The cashier's pay is by b each year, for 50 years. |
pay after 50 yr = 29{,}469 \(\cdot\, b^{50}\) |
| 2 |
To catch up, that has to equal the CEO's pay. Divide both sides by 29,469: pay grows by a of . |
\(b^{50} = \dfrac{27{,}400{,}000}{29{,}469} \approx\) |
| 3 |
The 14% row gives 700.23. The 15% row gives 1083.66. So b is 14% and 15%. |
\(1.14^{50} =\) and \(1.15^{50} =\) |
| 4 |
930 is much closer to 1083.66 than to 700.23, so b is 15% than to 14%. |
By trial: \(1.146^{50} \approx 925\), \(1.147^{50} \approx 970\) |
| 5 |
The exact answer is the on b that makes the equation true. The table gives an estimate of about . |
\(b \approx\) (i.e., about a 14.6% raise per year) |
Page 1 of 3 · What Is a Logarithm? (CEO Pay Catch-Up) · Algebra 2 · Unit 5 · Lesson 9
| Annual raise r | Yearly factor b | After 50 years b50 |
|---|
| 0% | 1.00 | 1.00 |
| 3% | 1.03 | 4.38 |
| 5% | 1.05 | 11.47 |
| 7% | 1.07 | 29.46 |
| 10% | 1.10 | 117.39 |
| 12% | 1.12 | 289.00 |
| 14% | 1.14 | 700.23 |
| 15% | 1.15 | 1083.66 |
| 20% | 1.20 | 9100.44 |
Worked Example #2 — Name the exponent: hello, logarithm
ProblemThe catch-up rate falls between rows of the table. To name the exact exponent, we use a logarithm: \(\log_{b}(x)\) is the exponent you put on b to get x. Rewrite each fact in log form, then use the table to read the exponent.
DataTwo key facts: \(\log_{2}(8) = 3\) because \(2^{3} = 8\). And from the catch-up table, \(1.03^{50} = 4.38\), so the same fact in log form names the exponent 50.
Word Bank
exponent
base
argument
value
multiply
power
subtract
Number Bank
2
3
50
4.38
1.03
10
2.8
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Start with \(2^{3}=8\). The 2 is the , the 3 is the , the 8 is the value. |
\(2^{3} = 8\) |
| 2 |
Rewrite as a log: "log base [base] of [value] = [exponent]." The exponent is . |
\(\log_{\,}\)\((8) =\) |
| 3 |
Same idea with the catch-up: \(1.03^{50}=4.38\). The base is , the value is . |
\(1.03^{50} = 4.38\) |
| 4 |
Rewrite as a log. The log gives the exponent, which is . |
\(\log_{1.03}(4.38) =\) |
| 5 |
Estimate: \(\log_{10}(630)\) is between which two whole numbers? \(10^{2}=100\), \(10^{3}=1000\). 630 is closer to 1000, so the answer is closer to 3 — about . |
\(\log_{10}(630) \approx\) |
Activity 2 — "Hello, Logarithm!" answers
1. \(\log_{10}(100) =\) 2. \(\log_{10}(1) =\) 3. \(\log_{10}(\,?\,) = 7\), so ? =
4. \(\log_{2}(8) =\) because \(2^{?} = 8\). 5. \(\log_{10}(630)\) is between and .
6. \(\log_{1.03}(4.38) =\) (read from the 3% row of the table).
Try saying: \(\log_{b}(x)\) asks "what on gives ?"
Agency Move — If 3% won't catch up, what would?
What rate would actually close the gap — and what would have to change to produce it?
The status-quo raise (3%) gets the cashier nowhere near the CEO in 50 years. The 50-year catch-up rate is about 14.6% — roughly 5 times the normal raise. Pick one real intervention and estimate its effect on b.
Status quo (given)
Raise: 3% (BLS 2024). \(b = 1.03\).
Catch-up: \(\log_{1.03}(930)\) \(\approx\) 231 years.
No working life is that long.
SK Hynix-style profit share
Effective raise jumps to ~10%. \(b = 1.10\).
Catch-up: \(\log_{1.10}(930) \approx\) years.
Still long, but inside a working life.
My pick: New b \(\approx\) New \(\log_{b}(930) \approx\) years.
Defend it: what real change would produce that rate, and who would have to do it?
Page 2 of 3 · What Is a Logarithm? (CEO Pay Catch-Up) · Algebra 2 · Unit 5 · Lesson 9
Tier 2 · General Academic Words
- compound — to grow on top of itself, so each year's growth adds to the next year's starting amount.
- raise — an increase in the amount of money someone is paid for their job.
- ratio — how many times bigger one number is than another (CEO pay is 930 times the cashier's, so the ratio is 930:1).
- gap — the space or difference between two amounts.
- percentage — a number out of 100 (a 3% raise is 3 dollars added for every 100 dollars of pay).
- salary — the amount of money a person is paid each year for their job.
- executive — a top boss who runs a company or a big part of one.
- decade — a time period of 10 years.
- estimate — a careful guess based on what you know.
- catch up — to grow until you reach the same level as something ahead of you.
- between — in the space that separates two values (the answer falls between rows of the table).
- argument — in math, the number you put inside the parentheses of a function.
- typical — the most common or average kind.
Tier 3 · Logarithm & CEO-Pay Words
- logarithm — the exponent you must put on a base to get a certain value. \(\log_{b}(x)\) is "the power that makes \(b^{?} = x\) true."
- base — the number being raised to a power. In \(\log_{b}(x)\), b is the base; in \(2^{3}=8\), 2 is the base.
- exponent — the power the base is raised to. A logarithm is an exponent.
- inverse — an operation that undoes another. The logarithm undoes raising to a power.
- compound interest — growth that multiplies each year, so the new growth is added to a bigger starting amount each time.
- CEO — Chief Executive Officer; the top boss of a company.
- board of directors — a small group of people who set big rules for a company and decide how much to pay the CEO.
- stock option — a deal that lets a person buy shares of a company at a low price later. CEOs are often paid mostly in stock options, which is why their pay can grow fast.
- executive compensation — all the pay a top boss gets: salary, bonus, and stock options together.
- median worker — the worker right in the middle of the company by pay. Half earn more, half earn less.
- pay ratio — the CEO's pay divided by the median worker's pay. A 930:1 ratio means the CEO earns 930 times as much.
- S&P 500 — a list of about 500 of the biggest U.S. public companies. People watch their CEO pay as a measure of pay across the economy.
- log base 2 — the logarithm that asks "what power on 2 gives this number?" \(\log_{2}(8) = 3\) because \(2^{3}=8\).
- profit-share — a deal where workers get a bonus equal to a percent of company profits.
- BLS / ECI — Bureau of Labor Statistics; the Employment Cost Index, which tracks how much average U.S. pay grows each year.
Page 3 of 3 · What Is a Logarithm? (CEO Pay Catch-Up) · Algebra 2 · Unit 5 · Lesson 9