Crown

Answer 1 (Logarithms):
I would describe the logarithm as the inverse operation of the power. Like subtracting is the inverse operation of adding, and dividing is the inverse operation of multiplying. I will start at the very low so it's hopefully a good guide to any level
The power (mathematically) is a short way of writing how many times one wants to multiply a number with itself. So if we have
2 * 2 * 2
we can also write it as
23
This saves time and paper (because maths people are lazy too ). When the line would go even longer, we save even more time. No-one wants to write or read or count a very long formula like below
3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3
It's shorter, easier, and more pleasing to write and read
315
A shout-out to the copy&paste function here. Using powers is very useful in this case. The "3" is called the base, the "15" is called the power (please correct me if I'm wrong).
Now let's to the reverse:
We have 8 as the answer and want to know the power. What has to be the "high" number to get eight?
2x = 8
This is where the logarithm comes into play. Like inverting an addition my subtracting. The logarithm gives us the power (literally) and it's written like this:
log2(8) = x
So here we have the "logarithm to the base two of eight". I think that's how it's pronounced, I very rarely "speak" math .
But the cool thing with this is that it can be done with any number. Almost. More below.
Most of the time when we talk about logarithms, it's the logarithm to the base of 10. I don't know why but that's what we do. Most calculators don't even give the option to enter a base for the logarithm. So the log button on the calculator is a log10 but with some maths magic it's possible to use a log10 to calculate the log for any base.
Onwards to the natural logarithm.
Or in short ln. Minor L, minor N. This one's is very special in many ways. It's the inverse operation to ex. That's the "natural" part in the "natural" logarithm.
e is is about 2.72 and is used to describe - or more-so describes - e.g. the natural decay of something. I fell like that's one of the very few occasions where maths meets the real world, waving through a thick window. So the natural logarithm is just the inverse operation of
ex
And because we need it quite often, it has his own name.
ln(16) = 2.773
The connection to KSP is almost anywhere where physics come in. In the rocket equation and in the atmosphere thickness. And even more but these are the ones I use quite often.
 
Here are some of the definition / rules about powers and logarithms:
n0 = 1
n1 = n
logn(0) = undefined
logn(b) is only defined for b greater 0.

 
I'll come back after breakfast. Back from breakfast.
Onwards to the Specific Impulse, Isp
Like @OhioBob already said, Specific Impulse is the engine efficiency and the exhaust velocity. Both. Together. Kinda. By mathematical magic.
I believe that the Isp is used in seconds because (besides of being shorter) it's unit - the second - is the only thing that American and German rocket engineers had in common. But I don't know if that's really true. But using seconds makes Russian and American rocket engines very easily comparable.
For myself, it was quite a challenge to wrap my head around the Isp but once it's in the head, it won't get out anymore.
On Wikipedia one can come across this neat (some would call it ugly) formula:

with
as the burn time,
as the fuel mass,
mean thrust,
thrust the the time ,
as the standard earth gravity.
You can ignore most of it.
A specific impulse of 1'000 m/s = 1'000 Ns/kg ≙ 102 s means that 1 kg of fuel can change the impulse by 1'000 Ns.
This means an engine that has a thrust of 1'000N and burns for 1 second uses 1 kg of fuel. Or a smaller engine with 10N of thrust that burns 10s and uses 0.1 kg of fuel.
1'000m/s is the change in speed the mass (1 kg) would experience if it would fall for 102s in nominal earth gravity.
Maybe it's easier to demonstrate the formula on an example. Let's take the beloved LV-T45 "Swivel" liquid fuel engine:
It has a vacuum thrust of 200kN of 200'000N and an vacuum Isp of 320 seconds.


When we put this in our neat formula, we get

With the units in bracket, it's not that confusing. But still, the newton is annoying, so ...

because 

 
Now the bad memories from 8th grade come back. But be can now kill almost any unit. the two s2 are eliminated, the meter m and the kg from the the newton and the mass. All what is left is the lonely second.
That's how we get the unit for the specific impulse. By killing units and leaving nothing more that the second alive.
 
But wait, there's more:
To get the exhaust velocity of the fuel multiply the Isp with the standard earth gravity. Use either Imperial or Metric:


 
Sometimes I want to know if I have enough fuel to perform a maneuver node burn. That's why I want to know the fuel consumption of the engine.
Let's take the formula and change it to our needs like Play-Do.

becomes
with the units        or   
With a burn time of 1 second, aka. fuel consumption per second

gives us 63.71 kg fuel per second. Divided by 5kg/unit we have 12.74 units fuel per second.
The fuel is mixed in a ratio (I hope it still is) of 11 Ox to 9 RP1, or 1.22. At this point I divide by 20 and multiply by 9 to get the RP1 consumption. That gives me 5.733 units RP1 per second. A FL-T400 with 180 units (liters?) of RP1 will last 31.39 seconds in vacuum with a LV45.
I use RP1 and Ox to have a linguistic difference between a tank's content ("fuel") and the resource ("fuel").
We could test our calculation by inserting the mass of 2'000kg and burn time of 31.30 seconds into the equation above. If we get an Isp of 320s, we are correct.
For my burn time calculations I usually ignore a few things. But I keep them in mind and in case something is odd I know where to look at. These things are the tb in the calculation above because it's 1 anyways. When it comes to the burn time for a delta-v, I ignore the whole basic rocket equation (e.g how long does it take to do a 850m/s burn and does my tank last this long?). This makes the burn a few seconds shorter and is good for my estimations.
I hope that was helpful in both mathematical and engineering ways. In case you have more questions, just ask ;-)
Lunch time!