How do you explain the concept of logarithm to a five year old? Okay, I understand that it cannot be explained to a 5 year old. But how do you explain the logarithm to primary school students?
 A: The base-$2$ logarithm of $64$ is how many times you have to multiply by $2$ in order to multiply by $64$.
$$
\underbrace{2\times2\times2\times2\times2\times2}_\text{There are 6 of these} = 64.
$$
$\log_2 64 = 6$.
NOTICE: "base-2" does NOT here refer to a base-2 numeral system, but to a base-2 logarithm.  All the excitement in the comments seems to have been about missing that point.
sigh......
A: The notion that a "general understanding of logarithms is out of the question at such a young age" is greatly skewed by our adult numerical perception. Much research suggests that children perceive magnitude through a logarithmically increasing function.1 A child's approach and performance in various estimation tasks shows that the shift towards a linear numerical conception occurs in stages. Preschoolers only linearize the numbers 0-10. Kindergarteners through 2nd graders linearize up to 100. From 2nd to 6th grade, the most significant change occurs: in studies involving tasks with the numbers 0-1000, 2nd graders consistently relied on a logarithmic model while 6th graders consistently relied on a linear model.2,3 Strikingly, indigenous peoples unexposed to a linear notion of numbers perceive quantity in an entirely logarithmic fashion at all ages.4
In a sense, Junior "knows" his stuff. He lives in his wild 'n cray logarithmic world where a unit measure isn't fixed. And by the way, you did too...
~:)   1                2               3              4         5      6   7 8 
------------------------------------------------------------------------------
:{⫐   1                2                 3                 4                 5 

...until you were violated by formal math. Forget your so-called "natural numbers." To teach junior we must first make sure we understand logarithms intuitively rather than computationally.
Get a "Feelin'" for the 'Rithm
Let's say I give you a cookie, and then I give you second cookie. How much more cookie do you have? "One cookie, clearly."
What if I give you a third cookie? How much more cookie do you have? "Again, one..."
Please humor me once more:
Is cookie three as much "cookie" as cookie two?
What if we add a fourth? If we try to feel "cookie" amount logarithmically, we can see a parallel with what we intuitively know as marginal utility. A young child sees all quantities in this relative fashion. (See the number line above.) A two person party feels like a huge leap from a party of one, but the jump from five to six just doesn't impress.
Even we slip up if we jump up a few magnitudes: \$1 to \$100? Woah buddy! \$1,000,000 to \$1,000,100? Psh.
So... how do you go about explaining logarithms to a child? With cookies of course! ;D
Totally Sweet Explanation of Logarithms
Let's say we have a baker, who makes cookies! This baker has a "powerful" oven. When the heat is low and he puts in one cookie, after one minute the cookie bursts into two cookies! Another minute later, each of those cookies burst into two again. With every minute, it happens again and again! So if the baker leaves his oven on for a long time, his whole bakery will fill with cookies! The baker can also set the oven to high. When he does this, each cookie in the oven turns to three after each minute. The baker uses those tools and to make numbers grow faster.  Just like the baker we have tools to make numbers grow very fast. The fastest is called the "power" and it works just like the oven, except with more notches  Instead of two or three cookies imagine five cookies popping up every time from every cookie! We can also put in more cookies at the beginning meaning even more at the end! If we keep using "power" our number gets bigger and bigger going, upwards until we can't even see how high it goes:

Now that baker... he loves cookies, especially his first. His second too, but not as much as the first. As he gets to his fifth he feels full, but man, those cookies are still pretty yummy, so he might have one more, but then after the hundredth cookie, the next cookie isn't that special, its almost the same as the ten before. Logarithms look the way the baker feels about eating that next cookie.  There's a big change in the beginning because the having the first after the second is really good, as but then the 101th is barely better than the 100th.  

After a while, the cookies aren't so much more exciting than the ones before. How far the hand reaches changes less and less. The cookies keep on growing more and more and more, but after a while we can't get any happier or excited than we already are. So they're a special type of opposites. We call this opposite of power/exponent in math a logarithm.
Now let's say the baker comes to our house and bakes cookies for us. After 10 minutes the whole house fills up with cookies! The baker can't remember what happened, so we investigate.  The baker had set the oven to a temperature, put some cookies in, and waited 10 minutes. Maybe he set the temperature too high. We go check the oven and see it set on low. That means he must have put in too many cookies! But how many cookies? Since we're trying to think backwards from the really fast growing of the cookies (in the power oven), we need to use the opposite way of thinking...  this opposite is called a logarithm.  If we count up all the cookies and use that number with the temperature and put it into the calculator, we can use a logarithm button to find out how many cookies the baker started with. Why do we need a calculator? Because sometimes there are just too many cookies. ;)
Footnotes:


*

*Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105(2), 395–438. doi:doi: 10.1016/j.cognition.2006.10.005

*Berteletti, I., Lucangeli, D., Piazza, M., Dehaene, S., & Zorzi, M. (2010). Numerical estimation in preschoolers. Developmental Psychology, 46(2), 545–551. doi:10.1037/a0017887

*Siegler, R. S., & Booth, J. L. (2004). Development of Numerical Estimation in Young Children. Child Development, 75(2), 428–444. Blackwell Publishing. doi:10.1111/j.1467-8624.2004.00684.x

*Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures. (2008). Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures. Science, 320(5880), 1217–1220. doi:10.1126/science.1156540
A: Put on a line, in equal distances, the numbers 1, 10, 100, 1000 and so on, and explain we wedge the other numbers in between, as appropriate (in a stretched way, like the powers of 10).
The base 10 logarithm is the distance from origin.
A: If you want to explain logarithms to primary school students, I think you'll want to teach them exponents first.
Try showing them a sequence of multiplied numbers, and telling them to count the occurrences:
\begin{equation}
    2_1 \cdot 2_2 \cdot 2_3 \cdot 2_4 \cdot 2_5
\end{equation}
The number of occurrences they count is the exponent, and that $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ is equal to   $2^5$
Then, show them that if they only know that 2 to the power of something was 32, they can use a log, where you put the base, like $\log_2$, and then the value, like $\log_232$, and that is equal to the number of occurrences.
I don't know if it's a good idea to go further than that with primary school students, but try it anyways.
Hope that helps.
A: Not all five-year-olds are alike, because some of them will go on to be physicists and biologists and others will be, say, driving a truck when they're 45.
Some forty-five-year-olds cannot understand logarithms, and some five-year-olds are going to be among those people in forty years.
On the other hand, a sufficiently bright five-year-old may well explain logarithms to you.
For the bulk of five year olds, it would probably be best to first teach them the concept of a function, and then the concept of a function's inverse. If the five-year-olds can grok functions and inverse functions, and the concept of exponentiation, then logarithms can be introduced as simply inverse functions of exponentiation: combine something you already know, with something else you already know.
A: The way I learned was pretty simple...
First, know exponents (repeated multiplication).
So, to what power do you have to raise the base to get the number?
This is the logarithm function.
A: Explain it using all the concets that is required, the age of student does not change the nature of logarithm, have it in notes that they can refer to time and time again. 
I have seen 6 years olds beat grown ups in chess, hell Maguns Carlsen was kikcing serious butt when was 10 and by 13 he was a Grandmaster (many adults try to get to that).
If the explantion is correct and complete then what difference does it make the reader is 5 or 50? Not all readers like the dumbed down version.
A: [I believe I described this idea in another answer --or maybe just a comment somewhere-- but I can't find it.]
Here's an exercise that might work with a group of kids. (Normally, I'd think they'd have to be older than 5, though.) I'll present this as if in a classroom.
Have the group single out the kid they consider the leader, hot stuff, numero uno; bring that kid forward and declare her "1" ... and also declare that all the other kids are just a bunch of zeroes. After they finish complaining that you insulted them, continue: "'1' is great, but she only represents herself." Call another kid --a zero-- forward. "'1', with one of you '0's, now represents ten. That's [say] the number of people in this room." Call up another kid. "'1' with two '0's is a hundred. That's the number of people on this floor of the building!" Another zero: "Four kids, and we've counted people in this building and a couple more the neighborhood!" Another zero: "This entire side of town!" More zeroes: "... this county! ... this state! ... this region! ... this country! ... this whole planet! ...! ...!! ...!!!" It's like the Powers of Ten movie, with kids acting as the zoom factor. If you like, blow the tykes' minds with the fact that you'll run out of representable particles in the observable universe before you run out of zero-kids in the whole school ---(Wikipedia says 80 kids will do.)--- imagine the size of the number represented if every kid on the planet got in line!
Anyway ... Once that's done, you can start asking this question: How many zero-kids do you have to line up to get a number representing as closely as possible this-or-that target number without going over. Powers of ten, of course, make this pretty easy to figure out if you just give the target's base-10 name; at the same time, powers of ten make for pretty rough estimates of numbers. So, up the challenge --and the accuracy-- by making each zero-kid represent, not a factor of ten, but, say, a factor of five or three or (ultimately) two.
At that point, you might stop calling the kids zeroes --unless you want to get into a discussion of numeral systems with different bases (and I'm not saying you don't or shouldn't want to do that)-- and call them something more (ahem) powerful, like "magnifiers" or "zoomers". (You might even do that from the very beginning, to avoid place value and notational distractions.) And the question remains: How many kids are needed to "magnify" or "zoom" the initial '1' into whatever target number? 
That "how many" is, of course, (the floor of) the logarithm.
If the OP can't find a whole group of precocious kids for this exercise, then just use legos or stuff animals with the single five-year-old. 
A: I think the easiest way to say it is "a log is a way to say how big a number is, when you don't want to say or write the number itself because it's so big". This is a working explanation that a 5-year-old can understand, given only two very basic things: numbers exist, and there are numbers higher than he knows how to count. It's also how we generally use the concept of logs as the order of magnitude; the exponent of a number in scientific notation is the integer log of the true value, and logs are used to make exponential growth look like linear growth, so we can compare very large numbers using relatively small ones.
If this is a very gifted child that knows how to multiply at 5 (most US kids don't practice that in school until about 3rd grade, but a bright second, maybe first-grader could get the concept if you illustrate it), then you could say that a log represents about how many times you would have to multiply 10 by itself to get the actual number. It would be easier to stick to whole-number logs for the first demonstration.
A: Why not use a  (eventually simplified or paper made) slide rule? Show the children how to multiply to get them interested and explain why it works (possibly measuring the lengths with an ordinary rule).
Let's elaborate a little on this (supposing that the children know well how to add and multiply) :  


*

*Build first two 1 meter (let's be generous... you'll be doing the job! :-)) 'linear' rules with regular graduations from 0 to 10 (and sub-graduations and...)

*Show the children how to add with these rules put side by side

*(possibly another time) ask them if they could multiply with these...

*at the appropriate instant show them your new super-slide-rule that is able to do multiplications (the length must be the same and graduations will be from 1 to 10)

*let them play with that...

*as an option, if you really want to introduce logarithms, you may propose using the first rule to measure the length from 1 to 2, 1 to 3 and so on... Proving that each time you add the length from 1 to 2 you get double the previous number (powers of 2) and in fact using the suggestions made by others here as long as they are not confusing for the children!


"Why, this is so simple a five-year-old child could understand it! Go find me a five- year-old child." Groucho Marx
A: You first show them what you do with multiplication:
\begin{equation}
    \underbrace{7+7+7}_\text{3 times} = 21,
\end{equation}
and you write $3\times7=21$.
Now when your operation is multiplication,
\begin{equation}
    \underbrace{7\times7\times7}_\text{3 times} = 343
\end{equation}
and you write $7^3=343$. Then number $3$ here is $\log_7343$.
A: Roughly speaking, a logarithm is like the the number of digits in a
decimal number.
A: When I was very young (under 10) my father taught me logarithms in base 10. 
The log of powers of ten was clear, and then he told me that interpolation was possible.
A practical application (which actually raised my questions) was chemistry. We were experimenting with acids and bases (and voltaic piles) and so understanding pH was a requirement.
Never underestimate the passion that a child can put in learning new fascinating things!
A: I would explain exponent first, however, not the traditional formula, but through combinatorics (i.e. the number of possible assignments) and then explain the inverse using the same situation (e.g. how many people are in the group, if each choosing from 3 possibilities results in 9, 27, 81 possibilities). If that goes alright, explain (as pointed by maurice) that it is close to the number of digits (each position in the number is a person that chooses from 10 digits), that logarighms work for real numbers as well, and that this way you can add numbers rather that multiplying them (again using the number of possible assignments; this may be a bit hard, but it is a great "discovery" when happens at last). 
This wasn't tested on primary schoolers, but I think it might be possible (I did understand logarithms at that age).
Hope that helps ;-)
A: Personally I always thought of a log that magically reversed $a^b = y$ so that $log_{a} y = b$ magically is given. Also after a few exercises of scaling using logs, which I find nice (the graphical representation) and the correlation between a power and a log graph, I got it down more firmly. Try showing with pictures? Surprisingly babies like logs :D
http://gaiamama.wordpress.com/2010/02/04/how-babies-count-facinating/
sorry, I read an article a few years back about how babies according to it prefer logarithmic counting and the traditional $1,2,3 ...$ is actually a departure from nature.
You can try with a number line but I think a graph would work better as a number line would not only quickly run out of numbers, but not show the relationship between the base and value it is being applied to. Use Log 2 and Log 10 first!
A: Repeated doubling is a good way to get big numbers fast.  Start with 1 and double it a few times: you get 1, 2, 4, 8, 16, ...  (You could tell them about bacteria dividing, if you want to motivate them to wash their hands; or use rabbits multiplying, but that maybe requires having another, even more challenging talk.)
How many times do you have to double 1 to get 32? 5 times.
How many times do you have to double 1 to get 128?  7 times.
How many times do you have to double 1 to get 50?  Well, 5 is not enough, but 6 is too many... we must be looking for a number between 5 and 6...
A: A logarithm is how much bigger or smaller a number is than another
number when you combine numbers using multiplication instead
of addition.
For instance, to see how much bigger $3$ is than $2$ in this way:
$$
\begin{array}{|cccccccccccccccc|}
\hline
2^1 & \cdot & 2^2 & \color{red}{2^3} & \cdot & 2^4 & \cdot & 2^5 & 2^6 & \cdot & 2^7 & \cdot & \color{red}{2^8} & 2^9 & \cdot & 2^{10} \\
\hline
 2  & \cdot &  4  &  \color{red}{8}  & \cdot & 16  & \cdot & 32  & 64  & \cdot & 128 & \cdot & \color{red}{256} & 512 & \cdot & 1024   \\
\cdot & 3   & \cdot & \cdot &  \color{red}{9}  & \cdot & 27  & \cdot & \cdot & 81  & \cdot & \color{red}{243} & \cdot & \cdot & 729 & \cdot \\
\hline
\cdot & 3^1 & \cdot & \cdot & \color{red}{3^2} & \cdot & 3^3 & \cdot & \cdot & 3^4 & \cdot & \color{red}{3^5} & \cdot & \cdot & 3^6 & \cdot \\
\hline
\end{array}
$$
This shows that
$$
1.5 = \frac{\color{red}{3}}{\color{red}{2}} < \log_2{3} < \frac{\color{red}{8}}{\color{red}{5}} = 1.6.
$$
You can measure $\log_2{3}$ as accurately as you like by comparing bigger powers
of $2$ and $3$.
(After this, either you end up with one very confused five-year-old, 
or else you've got a budding Eudoxus on your hands.)
A: First I'd explain exponential growth.  I'd give hands on, demonstrable examples like musical frequency and sound volume.  I'd demonstrate linear growth in sound, and linear growth in volume, and compare that to exponential.  Keep it concrete, it will be a challenge for most people, but fortunately learning to setup a demonstration like this will improve the instructor as well.
Then I would show how a logarithm is another way of looking at it (we perceive musical notes and volume in terms of their logarithm, which is how far along the exponential curve you are).
An alternative approach is to teach them how to multiply and divide and compute exponents using logarithm tables.  Most wouldn't care, but there are a few who would be interested to learn that calculators are not magic.  After showing them how to use the log tables, then perhaps use some algebra to show how it works.  Don't make the mistake of trying to teach the algebra first: children are not little Hilbert logic enumerators that extrapolate from proofs into real world application.  Show the useful part first.  If you don't know how to do arithmetic with log tables, grab any math textbook that was written between 1300 and 1940.
A: Take 2 and go on multiplying with the same 2. Can you add one to how many times you had to multiply to get to 32? That number is your log ( of 32 to base 2).  
A: I realize this is quite old a thread (7+yrs old!). I do think though I can provide a meaningful new perspective to the answers given so far. Let's see.
Short answer.
Present the kid with the following riddle (set up whatever preface you want for making the challenge more appealing to the kid): 

You a have bag of candies when you come across a friend who had a bad day and he's sad. To cheer him up you divide your amount of candies in two equal-sized halves and you give one half to your friend and keep the other. You keep on walking when you meet another friend. She's also sad. Again you divide the amount of candies you have in two halves and give one to this friend too. After a little while you find a third friend and you do the same. You keep doing the same with other friends until you have only one candy left. With how many friends can you split your candies?
  Answer: $lg_2(\#candies)$

Long answer.
I posted a question a little while ago that may help further clarify my point. See here.
If the kid is at least in grade 9, I've successfully used this approach to introduce the concept of a logarithm. A grade 12 student can easily use it to estimate arbitrary logarithms up to 1 decimal places.

$\log_b(x)\,\equiv\,$ the number of repeated divisions ($b>1$) / multiplications ($b<1)$ of $x$ by $b$ until you get a 1 

Examples:
1. $\log_58\,?\,Ans:8/5=1.6$ So the actual value is between $1$ and $2$, although closer to the first. For guessing a value, let's divide that interval in 4 parts.   As we expect it closer to 1 than to 2, we can pick a guess like $1.25$. The actual value is $1.29$.
2. $\log_{13}57\,?\,Ans:57/13\approx 4.4$, hence again the value of this log lies in $[1,2]$. As $4.4/13\approx 0.3$, we may guess the value as $1.3$. The actual value with one decimal is $1.6$.
3. We can do better by first enlarging the number. Example: $\ln 2\,?\,Ans:\mbox{Let's consider first say } \ln (2^{10})$. Now $1000$ can be divided by $3$ six times before the result is smaller than $1$: $1000/3^6\approx 1.37$. That's about half of $3$ and as $e<3$, we'll guess $\ln(1024)\approx 7$. Hence, $\ln(2)\approx 7/10=0.70$. The actual value is $\ln 2\approx 0.69$.   

A geometric interpretation has to do with zoom levels. Let's consider the following game. 

The rules are basically as follows. Two players compete against each other each in turns. Each turn is time constrained. Each player has access to a zoom-dial that controls the zoom level of a picture. Each dial works in a different scale. Say player 1's dial zooms in(turn clockwise)/out(turn counter-clockwise) by a factor of 2, while that of player 2 by a factor of 8. That information is known to the players. What a player does to the picture the other has to undo. The first player that fails to undo a change in the allotted time loses the game. 
Let's call the players Alice and Bob. One player, say Alice, starts by zooming in or out the picture using the dial she's been given. She tells Bob how many notches she turn hers -say $4.5$. Now Bob must undo that using his own dial. How much does Bob need to turn his dial to undo Alice's zoom? Answers: $4.5\cdot\log_{\mbox{Bob's zoom factor}}(\mbox{Alice's zoom factor})\,=\,4.5\cdot\log_8(2)\,=\,4.5/3\sim 1.5$.
Addendum:
What does $\log_{0.7}(2)$ mean? When the base is smaller than one, the log refers to the number of times we multiply $x$ by $b$ until we get $1$. The result is convened to be written negative to indicate that the process to get to $1$ is now through multiplications instead of divisions.
Summary:
The $\log_b x$ has to do with the number of steps needed to get to $1$ starting on $x$ with the only possible operations being multiplication  $*$ and division $/$ by $b$.
Conclusions:
Id' say that both pictures, the samaritan and the zooming challenge, may be grasped by at least a teenager (grade 7 on...not sure when they start learning division nowadays). 
The persistent problem when explaining logarithms is that no straight answer is taken, but instead a detour is taken by relating it to the exponential -worst is when referring to the log as the inverse of the exponential: Isn't there a (non-written?) rule that explanation of something should never be given in the form of "the opposite of" or "the negation of"? I, at least, feel that's true. The approach I present introduces an exponential only implicitly.
Additional details on this point of view of mine can be found here.
A: 
I recently proceeded as above with more than 5 year old students ( in fact 17-18 y. old). Simply I was surprised to see that they seemed to get it immediatly; which made me suppose that much younger students could grasp this explanation. 
I think that students already have a " logarithm  box" in their head. I mean, before having heard the term logarithm, they already have the concept of " the power to which I have to raise number $5$ to get number $25$", or "the power to which I have to raise number $2$ to get number $8$". 
If this  assumption  is correct, teaching what is a logarithm simply amounts to giving a name to this concept, to this " logarithm box". 
