How Logarithms Work
Demystify logarithms by understanding them as power counters and scale shifters. Learn the mental models that make calculation rules second nature while avoiding the algebra traps that trip everyone up.
Like Duolingo, but for How Logarithms Work. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
21 levels across 3 sections, about 42 minutes end to end, roughly 8 days at five minutes a day. It moves through Making the Concept Click, Rules and Common Traps, and Real Scales and Natural Growth. It starts from scratch.
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Key ideas in How Logarithms Work
- A logarithm calculates how many times to multiply the base by itself to hit the target value
- The base is what you repeatedly multiply (here, 2), and the output is the count of multiplications (5 times gives 32)
- Logarithms are not simple division or subtraction
- Exponentiation takes a starting recipe (base and number of steps) and gives the destination
- A logarithm starts at the destination and counts backwards to find how many steps of the base were used
- In log_b(x) = y, b is the base, y is the exponent, and x is the resulting product
- Both forms express that multiplying 5 by itself 3 times equals 125
- The translation loop starts at the small base at the bottom (4)
- The loop crosses the equals sign to pick up the power (3)
- The loop ends inside the log at the target value (= 64)
- Multiplication starts from a neutral baseline of 1
- Applying zero multiplications (power of 0) keeps any base at 1
- Therefore, log of 1 asks 'how many times do you multiply?' and the answer is always 0
- A base-10 logarithm counts how many times 10 multiplies together to reach a number
- log10(100) equals 2 and log10(1,000,000) equals 6 because of the number of zeros
- Moving up by 1 in log base 10 means multiplying the scale by 10
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives How Logarithms Work the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
What question does a logarithm answer?
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Where How Logarithms Work takes you
- 1
Making the Concept Click
- Asking the Reverse Question
- Base 10 and Base 2 in Real Life
- 2
Rules and Common Traps
- Addition and Multiplication Rules
- Handling Exponents and Negative Outputs
- Forbidden Inputs and Domain Traps
- 3
Real Scales and Natural Growth
- Reading Sensory and Physical Scales
- The Natural Logarithm and Continuous Change
3 sections · 7 units · 21 levels. Built to play, not to enroll.
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How Logarithms Work is taught in the The Professor style: clear, structured, thorough. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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