Logarithms
Slide along a log ruler where every step multiplies, and see why logarithms turn multiplying into adding.
Standards: F-LE.A.4F-BF.B.5
1Wonder
A magnitude 7 earthquake is not a little stronger than a magnitude 6: it shakes the ground 10 times as much. Sound in decibels and acidity in pH work the same way. These scales count “how many times we multiplied”, and that count is a logarithm.
2Explore
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Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.
- With base 2, press × b a few times. How does the number change? How does its logarithm change?
- Find log₂ 8 and log₂ 16. Then try base 10: what is log₁₀ 1000?
- Slide to a number between 1 and 2. Is its logarithm between 0 and 1?
- What is the logarithm of 1, whatever the base? Can you find the logarithm of a number less than 1?
3Explain
A logarithm asks “what power?”
2 to what power gives 8? Multiplying 2 three times gives 8, so the answer is 3. We write log₂ 8 = 3. A logarithm undoes an exponent.
The logarithm tells how many times you multiply the base b to reach x. It is the inverse of the exponential function.
4Practice
log₂ 16 = ?
5Recap
- log_b x is the power you raise b to in order to get x.
- log₁₀ of 10, 100, 1000 is 1, 2, 3; log_b 1 = 0 for any base.
- On a log scale, equal steps multiply.
- log(xy) = log x + log y: logarithms turn multiplying into adding.
- y = log_b x is the inverse of y = bˣ.