user1698: In the academic journal article "The Role of Deliberate Practice in Chess Expertise" from APPLIED COGNITIVE PSYCHOLOGY Appl. Cognit. Psychol. 19: 151– 165 (2005)
At the top of page 162, the authors state that

"...each log unit of serious study alone yields about 200 rating
  points compared to 33 rating points for log tournament play. Hence,
  players ought to devote the majority of their time to that activity."

I have gone over this academic paper several times, but I cannot decipher what "each log unit" represents. What time span is a "log unit" in this context where studying chess alone gave the chess players in this academic paper 200 ratings points vs 33 ratings points from playing in tournaments? Is it 6 months, 10 months, a year? 

user11245: The "log unit" term refers to a unit increment change in that variable, converted into the logarithmic equivalent.
The Table 3 on p. 158 shows how the variables that the researchers investigated are related to the output variable, which is ELO rating.
The variables included: 
v1: Hours of serious study
v2: Hours of tournament play
v3: Years of private instruction
v4: Years of group instruction
v5: Current hours/week serious study
v6: Current hours/week tournament play
Each of these is converted into a logarithm before the equation's factors are calculated. (The factor of a variable is the multiplier of that variable that produces the closest prediction to actual results, when all factors and variables are included.)
Their regression model shows that (if you use only these variables and the logarithmic approach) the peak ELO of a player will be best predicted by the equation:
PeakELO = 1145 +
195 x log(v1) + 32.7 x log(v2) +
9.4 x log(v3) +  4.3 x log(v4) +
6.3 x log(v5) + 20.3 x log(v6)

where 1145 is just a constant that helps improve the prediction.
They point out that the variables with the strongest effect on the prediction were those with the highest factors; v1 (total hours of serious study), and v2 (total hours of tournament play), but the first was much stronger than the second (195 vs 32.7, which is nearly 6 times as strong a factor), and dominated the other factors in their effect on the prediction.
The short answer to your question is that a log unit for the variable of total hours of serious study is an increment of 1, which transforms into different numbers of actual hours, depending on what the increment is being added to.

To illustrate what this means, if I've studied 1,000 total hours, then the log of 1000 = 3. My predicted PeakELO (for that variable alone) becomes
PeakELO = 1145 + 195 x log(1000) = 1145 + 195 x 3 = 1145 + 585 = 1730 ELO
(I'm ignoring the other variables for simplicity, but the results show that they're not substantial anyway.)
If I work 1 more hour, then 
PeakELO = 1145 + 195 x log(1001) = 1145 + 195 x 3.0004 = 1145 + 585.086
which is an indistinguishably small change.
If I work 100 more hours, though:
PeakELO = 1145 + 195 x log(1100) = 1145 + 195 x 3.041 = 1145 + 593 = 1738 ELO
So, while I'm at 1730, 100 hrs more of study will boost my ELO by 8 points.

But look at what happens if I've only studied 500 hours in total so far:
PeakELO = 1145 + 195 x log(500) = 1145 + 195 x 2.70 = 1145 + 526 = 1671.
Now, if I study an additional 100 hrs, 
PeakELO = 1145 + 195 x log(600) = 1145 + 195 x 2.78 = 1145 + 542 = 1687.
The same 100 hours boosts my ELO by 16 points.

What the formula is clearly doing is reflecting the diminishing returns of the same number of hours of study time as your ELO increases.  It accomplishes this accurate depiction of what actually happens by using the logarithm of the underlying data.
However, I would question the simple interpretation that the investigators offer, which is that a log unit increment of 1 results in 200 ELO increase.  Well, log of 1,000 is 3. Log of 10,000 hrs is 4.
So, they claim that I would get 200 ELO improvement if I had already studied 1,000 hours, and now committed to studying another 9,000 hours (yes, that's about 4.5 years of full-time study).  In their model, I would advance from 1730 ELO to 1930 ELO in no less than that time. 
Clearly, this cannot be true for every player.  I would suspect the following:
a) some confounding variables are having an effect, such as effectiveness of study (as determined by periodic assessment), ability to implement knowledge in tournament games (as determined by post-game analysis vs established knowledge base), age of player which the investigators acknowledge the model does not take into account, and so on...
b) the model's coefficient of correlation estimate (R^2) is only 0.34. Perfect correlation has 1.0; chaos (no correlation) is 0. So, 0.34 is promising, but not wholly effective by a long shot.  Also, the Standard Estimate of the Error is 222, meaning the predicted ELO could be off by over 200 ELO for a particular player, and it wouldn't be remarkable. That's a pretty wide margin.  So, in short, the model needs substantial improvement to be useful as a predictor of ELO.  
What the model does accomplish at the moment is to suggest that study is 6 times more effective than tournament play, which is in turn more effective than any other training method, including coaching, for the average player.

user291: The first paragraph of the RESULTS section on p. 155 says

We applied log-10 transformations to three variables (cumulative study alone, cumulative tournament play, and chess book library size) prior to correlation and regression analyses.

The log-10 (logarithm base 10) of a number is what power you have to raise 10 to in order to produce that number; e.g. log 100 = 2, log 1000 = 3, etc.
So the sentence

each log unit of serious study alone yields about 200 rating points

means that, all other things being equal, if player A studies ten times as long as player B, his rating will be about 200 points higher. (Another student who studies a hundred times as long as player B will have a rating of about 200 points higher than player A.) So there is no constant time span per 200-point gain; the required time to gain a particular amount of rating improvement gets larger and larger as you improve. (Another way to put this is that chess study produces diminishing returns.)
(In a sense, the Elo rating system is logarithmic as well, so you could claim that the relationship between study time and playing skill is actually linear. But since everybody measures playing skill by Elo rating, that's not so productive.)

1698: <p>In the academic journal article <strong>"The Role of Deliberate Practice in Chess Expertise"</strong> from <a href="http://www.psych.utoronto.ca/users/reingold/publications/PDFs/Charness.Tuffiash.Krampe.Krampe.Vasyukova.2005.pdf" rel="nofollow">APPLIED COGNITIVE PSYCHOLOGY Appl. Cognit. Psychol. 19: 151– 165 (2005)</a></p>

<p>At the top of <strong>page 162</strong>, the authors state that</p>

<blockquote>
  <p>"...each log unit of serious study alone yields about 200 rating
  points compared to 33 rating points for log tournament play. Hence,
  players ought to devote the majority of their time to that activity."</p>
</blockquote>

<p>I have gone over this academic paper several times, but I cannot decipher what <strong>"each log unit"</strong> represents. What time span is a "log unit" in this context where studying chess alone gave the chess players in this academic paper 200 ratings points vs 33 ratings points from playing in tournaments? Is it 6 months, 10 months, a year? </p>
 11245: <p>The "log unit" term refers to a unit increment change in that variable, converted into the logarithmic equivalent.</p>

<p>The Table 3 on p. 158 shows how the variables that the researchers investigated are <em>related</em> to the output variable, which is ELO rating.</p>

<p>The variables included: </p>

<p>v1: Hours of serious study</p>

<p>v2: Hours of tournament play</p>

<p>v3: Years of private instruction</p>

<p>v4: Years of group instruction</p>

<p>v5: Current hours/week serious study</p>

<p>v6: Current hours/week tournament play</p>

<p>Each of these is converted into a logarithm <em>before</em> the equation's factors are calculated. (The <em>factor</em> of a variable is the multiplier of that variable that produces the closest prediction to actual results, when all factors and variables are included.)</p>

<p>Their regression model shows that (if you use only these variables and the logarithmic approach) the peak ELO of a player will be best predicted by the equation:</p>

<p>PeakELO = 1145 +</p>

<pre><code>195 x log(v1) + 32.7 x log(v2) +
9.4 x log(v3) +  4.3 x log(v4) +
6.3 x log(v5) + 20.3 x log(v6)
</code></pre>

<p>where 1145 is just a constant that helps improve the prediction.</p>

<p>They point out that the variables with the <em>strongest effect</em> on the prediction were those with the highest factors; v1 (total hours of serious study), and v2 (total hours of tournament play), but the first was much stronger than the second (195 vs 32.7, which is nearly 6 times as strong a factor), and dominated the other factors in their effect on the prediction.</p>

<p>The short answer to your question is that a log unit <em>for the variable of total hours of serious study</em> is an increment of 1, which transforms into different numbers of <em>actual hours</em>, depending on what the increment is being added to.</p>

<hr>

<p>To illustrate what this means, if I've studied 1,000 total hours, then the log of 1000 = 3. My predicted PeakELO (for that variable alone) becomes</p>

<p>PeakELO = 1145 + 195 x log(1000) = 1145 + 195 x 3 = 1145 + 585 = 1730 ELO</p>

<p>(I'm ignoring the other variables for simplicity, but the results show that they're not substantial anyway.)</p>

<p>If I work 1 more hour, then </p>

<p>PeakELO = 1145 + 195 x log(1001) = 1145 + 195 x 3.0004 = 1145 + 585.086</p>

<p>which is an indistinguishably small change.</p>

<p>If I work <em>100</em> more hours, though:</p>

<p>PeakELO = 1145 + 195 x log(1100) = 1145 + 195 x 3.041 = 1145 + 593 = 1738 ELO</p>

<p>So, while I'm at 1730, 100 hrs more of study will boost my ELO by 8 points.</p>

<hr>

<p>But look at what happens if I've only studied 500 hours in total so far:</p>

<p>PeakELO = 1145 + 195 x log(500) = 1145 + 195 x 2.70 = 1145 + 526 = 1671.</p>

<p>Now, if I study an additional 100 hrs, </p>

<p>PeakELO = 1145 + 195 x log(600) = 1145 + 195 x 2.78 = 1145 + 542 = 1687.</p>

<p>The same 100 hours boosts my ELO by 16 points.</p>

<hr>

<p>What the formula is clearly doing is reflecting the <em>diminishing returns</em> of the same number of hours of study time as your ELO increases.  It accomplishes this accurate depiction of what actually happens by <em>using the logarithm of the underlying data</em>.</p>

<p>However, I would question the simple interpretation that the investigators offer, which is that a log unit increment of 1 results in 200 ELO increase.  Well, log of 1,000 is 3. Log of 10,000 hrs is 4.</p>

<p>So, they claim that I would get 200 ELO improvement if I had already studied 1,000 hours, and now committed to studying another 9,000 hours (yes, that's about 4.5 years of full-time study).  In their model, I would advance from 1730 ELO to 1930 ELO in no less than that time. </p>

<p>Clearly, this cannot be true for every player.  I would suspect the following:</p>

<p>a) some confounding variables are having an effect, such as <em>effectiveness of study</em> (as determined by periodic assessment), <em>ability to implement knowledge in tournament games</em> (as determined by post-game analysis vs established knowledge base), <em>age of player</em> which the investigators acknowledge the model does not take into account, and so on...</p>

<p>b) the model's coefficient of correlation estimate (R^2) is only 0.34. Perfect correlation has 1.0; chaos (no correlation) is 0. So, 0.34 is promising, but not wholly effective by a long shot.  Also, the Standard Estimate of the Error is 222, meaning the predicted ELO could be off by over 200 ELO for a particular player, and it wouldn't be remarkable. That's a pretty wide margin.  So, in short, the model needs <em>substantial improvement</em> to be useful as a predictor of ELO.  </p>

<p>What the model does accomplish at the moment is to suggest that <strong>study is 6 times more effective than tournament play, which is in turn more effective than any other training method, <em>including coaching</em>, for the average player</strong>.</p>
 291: <p>The first paragraph of the RESULTS section on p. 155 says</p>

<blockquote>
  <p>We applied log-10 transformations to three variables (cumulative study alone, cumulative tournament play, and chess book library size) prior to correlation and regression analyses.</p>
</blockquote>

<p>The log-10 (logarithm base 10) of a number is what power you have to raise 10 to in order to produce that number; e.g. log 100 = 2, log 1000 = 3, etc.</p>

<p>So the sentence</p>

<blockquote>
  <p>each log unit of serious study alone yields about 200 rating points</p>
</blockquote>

<p>means that, all other things being equal, if player A studies ten times as long as player B, his rating will be about 200 points higher. (Another student who studies a hundred times as long as player B will have a rating of about 200 points higher than player A.) So there is no constant time span per 200-point gain; the required time to gain a particular amount of rating improvement gets larger and larger as you improve. (Another way to put this is that chess study produces diminishing returns.)</p>

<p>(In a sense, the Elo rating system is logarithmic as well, so you could claim that the relationship between study time and playing skill is actually linear. But since everybody measures playing skill by Elo rating, that's not so productive.)</p>
