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?
Roughly speaking, a logarithm is like the the number of digits in a
decimal number.
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 ($b&gt;1$) / multiplications ($b&lt;1)$ 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 $e&lt;3$, 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.