Okay I found one of the notebooks I had some of this stuff in. Here's a quick way to approximate logarithms (base 10) but I don't have how accurate this is in terms of %'s but it should be pretty damn close. The key is that you just need to memorize the approximations for the log of 1-9 if you haven't already. Here is a list and a 3x3 matrix of the results that I can still remember them by even though I haven't used this in over a year.

log 1 = 0
log 2 = 0.3
log 3 = 0.5
log 4 = 0.6
log 5 = 0.7
log 6 = 0.8
log 7 = 0.85
log 8 = 0.9
log 9 = 0.95

Code:
0     0.3  0.5
0.6   0.7  0.8
0.85  0.9  0.95
And for completeness, log 10 = 1.

For whatever reason I can remember this 3x3 by rows pretty easily so maybe you can too. Now that you know this, doing logs in your head is pretty easy as long as you know [remember] how to manipulate logs in the first place. Here are a few guidelines that you learn in algebra or pre-Calculus or where ever it's taught these days:

log(a*b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b * log(a)

So here are a few quick examples.

Example 1: Find log(28).

Answer: log(28) = log(4*7) = log(4)+log(7) = 0.6 + 0.85 = 1.45.
(The real result is ~1.447).

Example 2: Find log(3000).

Answer: log(3000) = log(3 * 10^3) = log(3) + log(10^3) = log(3) + 3 log (10) = 0.5 + 3 = 3.5.
(The real result is ~3.477).

Example 3: log(36) = log(6^2) = 2 log(6) = 2 * 0.8 = 1.6.
(The real result is ~1.556).

So there you go. Maybe I'll do some trig stuff at another time.