Topics Numbers and logic

What is a logarithm?

A logarithm is the answer to the question: what exponent do I need? If 10 raised to the power of 3 is 1,000, then the base-10 logarithm of 1,000 is 3. It is exponentiation run backward, the way subtraction undoes addition or division undoes multiplication.

What makes logarithms interesting is what they do to scale. They turn multiplying into adding, which is why people used them to do big calculations by hand for centuries, and why slide rules worked. They also squeeze huge ranges into small ones, which is why earthquake magnitude, sound loudness in decibels and acidity in pH are all measured on logarithmic scales. Each step up means a multiplication, not an addition.

An episode on this would start from a picture instead of a formula: counting how many times you can double something. From there it would cover bases, the number e, and where logs appear in real life. bre's hosts are AI, so they can be wrong, and you can press Talk to ask them to slow down or try another example.

What a bre episode would cover

An outline of the episode bre would make for this question. Every episode is written fresh when you ask, so yours will differ.

  1. The question a logarithm asksStart with powers: 2 times itself three times is 8. A logarithm flips that around and asks how many times you multiply to land on a given number.
  2. Why multiplying becomes addingExponents add when you multiply powers of the same base, so logs turn multiplication into addition. This is the trick behind logarithm tables and slide rules.
  3. Bases: 10, 2 and eBase 10 suits our number system, base 2 suits computers, and e, about 2.718, shows up naturally in growth and calculus. Changing between bases is a simple conversion.
  4. Logarithmic scales in the real worldDecibels, the Richter scale and pH all use logs, so each step is a multiple, not a fixed amount. That is why a small change on the scale can mean a large change in reality.
  5. Logs and exponential growthWhen something doubles over and over, a log tells you how long it takes to reach a target. It also turns a curving growth chart into a straight line.
  6. Where logs hide in computingSearching a sorted list by halving it takes about as many steps as the base-2 log of its size. That is why some algorithms stay fast even as data gets huge.

How the episode might open

A sample exchange between two of bre’s AI hosts, bre and Arlo. Both are AI; this is written by AI, as every bre episode is.

  1. breAI host

    Okay, a question I think a lot of people quietly carry from school: what actually is a logarithm? Because I remember the button on the calculator, and I remember not knowing why it was there.

  2. ArloAI host

    It's a backward power.

  3. breAI host

    That's the short version, and I'll let you have it. But say it with numbers. If I have 2 times 2 times 2, I get 8. So what's the logarithm doing there?

  4. ArloAI host

    Takes the 8, gives you the 3. Base 2. How many twos did you multiply? Three. That's the log.

  5. breAI host

    So it counts the multiplications. Not the number itself, but how many steps it took to get there.

  6. ArloAI host

    Yep. Counts the steps. Same idea, flat.

  7. breAI host

    Here's what I want to know, though. Why would anyone bother counting steps instead of just using the number?

  8. ArloAI host

    Because the numbers get stupid big. Steps stay small. Who wants to read a million zeros?

Questions people also ask

What does log mean in math?
Log is short for logarithm. The expression log base b of x asks what power you raise b to in order to get x. For example, log base 2 of 8 is 3, because 2 to the third power equals 8.
What is the difference between log and ln?
Log usually means a logarithm with a stated base, often 10 on calculators. Ln means the natural logarithm, which uses the number e, about 2.718, as its base. Some books and programming languages use log to mean the natural one, so check the context.
Why are logarithms used for earthquakes and sound?
Those quantities span enormous ranges, so a log scale compresses them into manageable numbers. On the Richter and decibel scales, each step up represents a multiplication of the underlying measurement, not a fixed addition. Exact factors depend on the scale.
Can you take the logarithm of zero or a negative number?
Not with ordinary real numbers. No power of a positive base ever gives zero or a negative result, so the log has no answer. Mathematicians can extend logs to negative numbers using complex numbers, but that is a separate topic.

Related topics

More: all 300 topics, numbers and logic, or the longer reads on /learn.

bre’s hosts are AI, and every episode is generated, so they can be wrong: check anything that matters. This page outlines what an episode would cover. It is for interest and learning, not medical, financial or legal advice.