Logs look scary. But a power and a log are the same machine, just run in opposite directions.
Part one: the power machine
This is the times-2 machine: whatever goes in comes out doubled. Put in 1 and run it three times: 2, 4, 8.
That's 2³ = 8. The power is how many times the machine runs.
Part two: the log machine
The log machine runs the other way. Feed it the 8 that came out, and it undoes the doubling: 4, 2, 1.
It took 3 runs to get back to 1. So log₂ 8 = 3.
How to read a log
- The small number (the base) tells you which machine.
- The big number is what came out.
- The log is how many runs.
Worked example: log₃ 81
Use the times-3 machine: 3, 9, 27, 81. That's 4 runs, so log₃ 81 = 4, because 3⁴ = 81.
Formally, logb x = n means bⁿ = x. Not every log is a whole number (log₂ 10 is about 3.32), but the machine is the right first picture.
What's log₂ 32?
Show the answer
Run the times-2 machine: 2, 4, 8, 16, 32. That's five runs, so log₂ 32 = 5.
Save this for your next test
- A power says how many times the machine runs.
- A log runs the machine backwards and counts the runs.
- The base tells you which machine.
- log₂ 8 = 3 because 2³ = 8.
Read the video transcript
- Logs look scary. But a log is just a machine running backwards.
- Part one: the power machine.
- This is the times-2 machine. Whatever goes in comes out doubled.
- Put in 1 and run it three times: 2, 4, 8.
- That's 2 to the power 3. The power is how many times the machine runs.
- Part two: the log machine. It runs the other way.
- Feed it the 8 that came out. It undoes the doubling: 4, 2, 1.
- It took 3 runs to get back to 1. So log base 2 of 8 is 3.
- The small number tells you which machine. The big number is what came out. The log is how many runs.
- Try one: log base 3 of 81. Use the times-3 machine.
- 3, 9, 27, 81. That's 4 runs. So log base 3 of 81 is 4.
- Now you: what's log base 2 of 32? Tell me in the comments.
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