# How best to explain the logarithm to the mathematically naive?

Suppose you need to explain "What is a logarithm?" to an intelligent but math-phobic adult (or a student well-prior to her introduction to logarithms).1 I have used base-$10$, saying that, essentially, the logarithm (base-$10$) of a number is the number of digits needed to represent it in decimal notation. But that's not quite right at the $10^k$ transitions. And neither floor nor ceiling makes a clean statement about the number of digits:

$$\lfloor \log_{10}(999) \rfloor = \lfloor 2.9996 \rfloor = 2$$ $$\lfloor \log_{10}(1000) \rfloor = \lfloor 3 \rfloor = 3$$ $$\lfloor \log_{10}(1001) \rfloor = \lfloor 3.0004 \rfloor = 3$$
$$\lceil \log_{10}(999) \rceil = \lceil 2.9996 \rceil = 3$$ $$\lceil \log_{10}(1000) \rceil = \lceil 3 \rceil = 3$$ $$\lceil \log_{10}(1001) \rceil = \lceil 3.0004 \rceil = 4$$
Is there some clean way to interpret $\log_{10}$ as the number of digits, without getting hung up on the detail that $1000$ needs $4$ digits but $\log_{10}(1000)=3$, or trying to explain scientific notation (which accords with the floor):
$$999=9.99\times10^2$$ $$1000=1.000\times10^3$$ $$1001=1.001\times10^3$$
The caveats I find necessary to explain the nuances destroy the clarity I seek to achieve.
1 In response to a request to explain what I mean by mathematically naive, I answered: "I was thinking of a generic family gathering. So in this context, most people do not understand exponents. But everyone understands that 1,234 has 4 digits."

• Incidentally: Another log question (not considered "related" by the site algorithm) was MESE 1820: Why aren't logarithms introduced earlier? – Benjamin Dickman May 25 '15 at 0:11
• What's wrong with saying the floor of the logarithm is the number of digits after the first digit? Or the number of spaces the decimal point needs to move to be after the first digit (which works with negative logarithms)? – Rory Daulton May 25 '15 at 0:43
• @RoryDaulton: Good. Essentially scientific notation but without heading into that thicket explicitly. – Joseph O'Rourke May 25 '15 at 1:08
• One way to explain it is by analogy with real-life phenomena. Earthquakes are measured on a logarithmic scale so a 7.0 is 10 times stronger than a 6.0, etc. Logarithms in practice are often used for linearlization like this and I think it's best served this way. – Cameron Williams May 25 '15 at 18:13
• Vi Hart has a pretty good video explaining logs: youtube.com/watch?v=N-7tcTIrers. There's a little bit of build up, but logs come in at about 6 minutes in. – Aeryk May 26 '15 at 15:48

What's wrong with

$\log_{10}(10)=1$

$\log_{10}(100)=2$

$\log_{10}(1000)=3$

$\log_{10}(10000)=4$

and explaining that log is the power to raise 10 to get a given number. I think the digit count is an unnecessary tangent to the process. The above 4 equalities lend them selves to showing how 1000/10=100 and for their logs, 3-1=2, etc.

On re-reading your question, for integer results, the base 10 log is the number of zeroes after the 1.

I've been thinking about your phrase "mathematically naive" and who might be in this group. Are these teens who are not quite at the level they should be, or adults who are not math savvy for whatever reason asking you to explain logs? What I'm trying to parse out here is what is the starting point? Do they understand exponents at all?

• I was thinking of a generic family gathering. So in this context, most people do not understand exponents. But everyone understands that 1,234 has 4 digits. – Joseph O'Rourke May 25 '15 at 13:29
• Since the OP speaks apparently about my wife, this is exactly what I did. – Claude Leibovici May 31 '15 at 15:37

If I where given some time, I would try to explain exponents and then explain logarithms, because it really explains what they are. In more details, one could say the following.

"Remember how multiplication is defined? It is just iterated addition: $3\times 4$ is defined as $4+4+4$, the sum of three fours. You learned that years ago, and then you learned how to interpolate so as to define the product of non-whole numbers, e.g. $1.2 \times 3.5$. You also learned how to divide numbers, and that $12/4$ is the number by which one multiplies $4$ to get $12$, i.e. $3$.

You may also have learned exponents; usually is not remembered that well, but it is really just the same process applied to multiplication: $4^3$ is $4\times 4\times 4$, i.e. the product of three fours. It is also possible to interpolate this operation to define it for non-whole numbers, e.g. $1.03^{6.5}$. This is a very useful operation, as the last example is the number by which your savings are multiplied in six and a half years if you have a $3\%$ interest rate. So, exponentiation is to multiplication what multiplication is to addition.

Now, a legitimate question is: what is the analogue to division for exponentiation? I.e., is there a way to find to what powers one must raise, say $10$, to get a given number, say $120$? Yes there is one, and this is, by its very definition, what a logarithm is. The base-$10$ log of $120$, often denoted by $\log(120)$ of the barbaric $\log_{10}(120)$, is precisely the number by which one has to raise $10$ to get $120$. It is even easy to have an idea of what its value is, because $100$ is $10$ to the power $2$, and $1000$ is $10$ to the power $3$, so that to get $120$ one has to raise $10$ to a power a little bigger than $2$. This operation is really almost as simple conceptually as division, but is more difficult to compute by hand."

• Very nice: exponentiation :: multiplication :: addition. – Joseph O'Rourke May 25 '15 at 20:59

I would not worry about an exact correspondence, because such an exact correspondence is not needed. There is a good but not exact correlation between the function that is logarithm to the base 10 of a positive real number and the function that counts digits of a positive integer, and the two functions are used for different purposes.

For such an introductory discussion, you can talk about number of digits, because that converts things from a scale that is hard to imagine physically to one that is easy to work with. Instead of writing something like the number of atoms in a grain of sand, the number of grains of sands on a planet, the number of planets in the universe, or the inverse ratio such as the size of a subatomic particle compared to the size of a galaxy, where such numbers are hard to get the number of digits right, much less the exact value, logarithmic approximations are easier to use. Even if they don't have the exact definition of logarithm at hand, the fact that they can count decimal digits, do addition and subtraction, and get close to the right answer should do for an introduction.

Gerhard "There's No Test Afterwards, Right?" Paseman, 2015.05.25

A simple answer is that it is calculating device to reduce multiplication to addition.

What about coming at it from a historical approach.

• How about making a table of logarithms? This should be within the capabilities of a 12 year old. This can be done using Excel pretty easily. There are some good resources I've found on the net.

• How about learning to use Napier's Bones? You'll probably need to make your own.

You are close.

Floor($\log_{10}(N))+1$ is the number of digits.

• How does that help explain a logarithm to the naive? This is better served as a comment than an answer. – Chris C May 26 '15 at 0:28
• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – Chris C May 26 '15 at 0:29
• Read the question...it asks if "there is a clean way to intepret log10 as the number of digits." – user52817 May 26 '15 at 3:40
• The question is how best to explain logarithms in the title with number of digits as a method. – Chris C May 26 '15 at 15:37
• $Log_{10}(N)$=Integer part+ decimal part=Characteristic + Mantissa. The characteristic encodes the number of digits needed for N. See above formula. The mantissa encodes the actual digits. A table of logarithms is used decode the digits from the Mantissa. Example: N=12340. $log_{N}(12340)=4+.0913$, 12340 has 4+1 digits, and then we find 0913 in a log table to reveal the digits 1234. Since N has 5 digits, we reconstruct that N=12340. – user52817 May 26 '15 at 19:32

\begin{align} \log_3 81 & = \log_3(3\times3\times3\times3) = 4 \\ & = \text{the number of multiplications by 3 that amount to a multiplication by 81}. \\[10pt] \log_3 1 & = 0 = \text{the number of multiplications by 3 that amount to a multiplication by 1}. \end{align}

If you are dealing with those who are naive, I would consider to just give them situational awareness. One of the things that is frustrating about Wikipedia, is if I want to learn ABOUT a math topic, not learn THE math topic, I get very little out of the articles. This is because math types devolve to full proof or rigorous definition (worst) or teaching the topic itself (better, but still too intricate).

Instead if you are dealing with naive people, just give them information to help them classify it within overall knowledge (not to learn the logarithms themselves...if you want to teach logarithms, than take algebra 2...and follow the pedagogical sequence therein.) So some things ABOUT logarithms, sans learning them, sans defining them:

*what course taught in: algebra 2, typically 11th grade.

*what later courses require it: calculus, chemistry, physics. Don't say bio (don't need it for freshman bio). Don't say differential equations. "Calculus" is enough...big hard course they have heard of but not taken.

*what related to: "like powers of stuff, you know squaring and cubings, but on steroids. Not the same thing as powers, but you get to logarithms after mastering your powers." [This assumes they have heard of the powers, even if not knowing exponent rules]

*useful for: helps to do complicated arithmetic. Calculators use it to do their calculations of big arithmetic stuff.

*involved with: pH in chemistry (how acid stuff is), Richter scale for earthquakes, sound level ("decibels"). [RESIST the impulse to lecture about nonlinearity...you basically have to explain what a logarithm is to do that...and for God's sake don't use the word "nonlinear" with the peeps. It's enough to just give them some motivational touchpoints that the topic relates to, that they have heard about.]

*funky looking graph paper: hand each student a page of log-log and of semi-log graph paper. Don't make it yourself. Go to the specialty engineering store and buy a pad of each paper. Nice and green and professional looking...you practically feel crew cutted 1950s engineers sending rockets into space from the paper itself. Let the audience keep the pages you give them. Resist any urge to run an exercise or graph something (too hard, benefits from knowing logs first). But you can say "look at that funky scale with the 10, 100, 1000, etc. and missing the zero". "Once you learn what logs are, that weird scale will make sense."

*Consider to make a joke or pun or two. Icebreaker and tension reliever. Something that does NOT require understanding what a log is. Something like "it's different from Lincoln's cabin type of log" (look it up, sure there is some site that has math puns on it...GIYF.)

Feel free to copy and paste my rewrite of these 2 r/ELI5 posts.

By user cambridge_ms, 2013 Nov 15

When you multiply a number by itself a few times, you can express that number in terms of an exponent. The exponent is a little number to the upper right of the number that says, "this is how many times the number has been multiplied by itself". So, $$2 \times 2 \times 2 = 2^3$$. The number 2 has the exponent 3.

The "logarithm" is the reverse of this operation. When we ask, "what is $$\log_28$$?" we are asking, "what is the base 2 logarithm of the number 8", or, "how many times did we multiply 2 (the base) to get the number 8". The answer to this question is the exponent from above.

Since $$2 \times 2 \times 2 = 2^3 = 8$$ (that is, the base 2 multiplied by itself 3 times equals base 2 to the exponent 3 power equals 8), we know that the $$\log_2(8) = 3$$. How many times did we multiply 2 to get the number 8? 3 times.

What this means, is that we can use logarithms to express HUGE numbers relative to each other in a much smaller space.

Imagine a graph that graphs data ranging from 2 to the number 1,048,576. That's going to be a graph with an enormous Y-axis scale (1 million is way bigger than 2).

However, if we took the base-2 logarithm of every data point, we'd end up with a y-axis that only goes from 1 to 20 ($$\log_2(2) = 1; \log_2(1048576) = 20$$). Thus, we often display data using a log scale because it allows us to relate HUGE numbers to each other on a scale that is much easier to understand.

By TheBB, Nov 8 2012

Suppose you measure something. By its very nature, this thing that you measure produces only positive numbers: this is a requirement to use a logarithmic scale. You get these measurements:

1, 3, 500 000, 0.00005

Ouch. That's quite some variation there. If you plot this on a normal scale, these measurements will look unsightly. One point (500 000) is way off somewhere, and the rest sort of clustered around zero. It'll be impossible for you to look at this plot and appreciate that there's a huge difference between 1 and 0.00005.

This is not a contrived example. I made it up, but lots of natural phenomena are like this.

So what do we do? Use a logarithmic scale! Take the logarithm of those numbers above: $$0, 0.477, 5.699, -4.301$$

If you plot these smaller numbers on a normal scale, things are much clearer. There's a tiny one, a huge one, and two middling ones.