Day 7 — Exponents And Logarithms
Time: ~30 min · Topic: exponents and logarithms
The problem
Solve for x: log_2(x) + log_2(x - 2) = 3
Hint: Remember the log rule: log(a) + log(b) = log(a * b). Once you combine them, think about what power of 2 gives you the right-hand side.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Use the product rule of logarithms to combine: log_2(x(x - 2)) = 3
- Convert from logarithmic to exponential form: x(x - 2) = 2^3 = 8
- Expand and solve the quadratic: x^2 - 2x - 8 = 0, which factors as (x - 4)(x + 2) = 0, giving x = 4 or x = -2
- Check the domain — logarithms require positive arguments, so x > 2. Reject x = -2. The answer is x = 4.
Answer: x = 4
Why this matters
Logarithms were invented in the 1600s by John Napier specifically to make multiplication easier for astronomers — they literally turned multiplication into addition, centuries before calculators existed!
This day in math: The King of Invariant Theory Is Born
April 27, 1837 · Paul Gordan
On April 27, 1837, German mathematician Paul Gordan was born in Breslau. Known as "The King of Invariant Theory," Gordan proved that the ring of invariants of binary forms is finitely generated — a landmark result in algebra. He also mentored Emmy Noether, one of the greatest mathematicians of all time, supervising her doctoral thesis at the University of Erlangen in 1907.
Why it matters: Gordan's work laid the foundation for modern abstract algebra, and his famous clash with Hilbert over constructive vs. existential proofs shaped how mathematicians think about what counts as a valid proof to this day. His mentorship of Emmy Noether helped launch a revolution in algebra and physics.
Did you know?
Zero factorial (0!) equals exactly 1 — not zero. This means that the number of ways to arrange absolutely nothing is precisely one way: do nothing at all. It's one of the most counterintuitive results in all of mathematics, and it's essential for countless formulas to work correctly.
Factorial means multiplying all positive integers down to 1, so 3! = 3 x 2 x 1 = 6. But there's a beautiful pattern: 4! = 24, 3! = 24/4 = 6, 2! = 6/3 = 2, 1! = 2/2 = 1... and following that same logic, 0! = 1/1 = 1. Defining 0! = 1 also makes the binomial coefficient and combinations formulas work perfectly — without it, expressions like "choose 0 items from n" would break. It's not just a convention; the math demands it.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact