The magnitude limit formula, L_{mag} = 2 + 5 log(D_{O}) appears to be
something that you can't simply calculate in your head. But... if you can find
log(D_{O}) in your head to within a tenth (i.e. within 0.1) you can then
estimate the magnitude limit within a half-magnitude.

Turns out this is something you __can__ do, usually in less than 15 seconds. Amuse your
friends! Amaze your pets! Baffle your colleagues! Best of all... know what to expect
from your telescope...

- Fundamentals of Logarithms
- What's a Log?

- The logarithm of a number is the exponent to which 10 is raised to produce that number.
- The logarithm of a number is written log(number)

- Integer Examples
- log(10) = 1, since 10
^{1}= 10 - log(100) = 2, since 10
^{2}= 100 - log(1000) = 3, since 10
^{3}= 1000 - log(1) = 0, since 10
^{0}= 1

- log(10) = 1, since 10
- Non-integer Examples

Non-integer logarithms result when the number is not an even power of 10- log(2) = 0.3
- log(3) = 0.477

- Operations with Logarithms
- When you multiply two numbers to get a result, you add their logarithms to get the logarithm of the result.
- Obvious example: 10*10 = 100; log(10) = 1, and 1+1 = 2, which is log(100)
- Less obvious example: 2*3 = 6; so log(2)+log(3) = 0.3+0.477 = 0.777 = log(6)
- When you divide two numbers to get a result, you subtract their logarithms to get the logarithm of the result.
- Obvious example: 1000÷10 = 100; log(1000) - log(10) = 3-1 = 2, which is log(100)
- Less obvious example: 10÷2 = 5; log(10) - log(2) = 1 - 0.3 = 0.7, which is log(5)

- What's a Log?
- How to Do Them in Your Head (notice there are two important tricks)

**Important Trick #1:**Round the Number to One Significant Digit

For example:- 76.2 becomes 80
- 355.6 becomes 400
- 635.0 becomes 600
- 2,540 becomes 3,000

- Find the log of the significant digit: this will be the decimal part of the log,
and sometimes is called the "mantissa".

*We only need this to the nearest tenth (0.1).* - Count the zeroes: this will be the integer part of the log
- Add the integer part and the decimal part and you're done.

- Knowing the Logs of the Single Digits

**Important Trick #2:**Two Logs to Remember- log(2) = 0.3
- log(3) = 0.5 (actually 0.477, and 0.5 is good enough)

- The rest can be found from those two, as explained in the table below.
Number Finding Log 1 **0**by definition2 **0.3**3 **0.5**4 2×2 ==> 0.3+0.3 = **0.6**5 10÷2 ==> 1.0-0.3 = **0.7**6 2×3 ==> 0.3+0.5 = **0.8**7 very close to 6, so we will call it same as 6 ( **0.8**)8 2×2×2 ==> 0.3+0.3+0.3 = **0.9**9 3×3 ==> 0.5+0.5 = **1.0**(it's so close, call it that)10 **1**by definition100 **2**by definition1000 **3**by definition - When the significant digit of your number (the mantissa) wants to fall
between 1 and 2, you can add a refinement. As you go from 1 to 2, you
jump from a logarithm of 0.0 to 0.3. If you want accuracy to 0.1, it's
a good idea to remember that
- 1.3 is a third of the way to 2, so its log is 0.1
- 1.7 is two-thirds of the way to 2, so its log is 0.2
- so if you have a number between 1 and 2, see if it's closer to 1.0, 1.3, 1.7, or 2.0.
- Then pick 0.0, 0.1, 0.2, or 0.3. Extreme precision is not required, just figure roughly which is closest.

- You can likewise assign values that are close to 2.5 a log of 0.4, again with the goal to get yourself to the nearest tenth, since going from a digit of 2 to 3 jumps you from a log of 0.3 to 0.5.

Once you master the above, examples are ridiculously easy. The hardest part is remembering to multiply by 5 and add 2 to get the magnitude limit of a scope.

Let's use the four scopes that are described on the How to Size Up & Set Up a Scope page.

- Meade ETX-80BB Backpack Observatory Telescope: D
_{O}= 80mm

You think:- 80 already has only one significant digit
- 8 is 2×2×2 so the log of 8 is 0.3+0.3+0.3 = 0.9
- ...and there's one zero so the integer part is 1
- So the log of 80 is 1 + 0.9 = 1.9
- oh, crud - I have to multiply by 5 and add 2, that's the hard part...
- Ok, 5 times 1.9 is 5*1 + 5*0.9 = 5 + 4.5 = 9.5
- Oh, then I have to add 2... so it's 11.5.

- Celestron AstroMaster 114 EQ Reflector Telescope: D
_{O}= 114mm

You think:- 114 has three significant digits, there's only room in my head for one...
so call it 100.

- log of 100 is 2
- Ok, 5 times 2 is 10 plus 2 is 12

- 114 has three significant digits, there's only room in my head for one...
so call it 100.
- Orion SkyQuest XT8 Classic Dobsonian Telescope: D
_{O}= 203mm

You think:- 203 is going to be 200 to get it to one significant digit
- log of 2 is 0.3
- ...and there are two zeroes so the integer part is 2
- So the log of 200 is 2 + 0.3 = 2.3
- oh, yuck - I have to multiply 2.3 by 5 and add 2...
- Ok, 5 times 2.3 is 5*2 + 5*0.3 = 10 + 1.5 = 11.5
- Oh, then I have to add 2... so it's 13.5.

- Obsession Telescopes 18-inch Dobsonian: D
_{O}= 457mm

You think:- 457 is going to be 500 to get it to one significant digit
- 5 is 10÷2 so log of 5 is 1.0-0.3 = 0.7
- ...and there are two zeroes so the integer part is 2
- So the log of 500 is 2 + 0.7 = 2.7
- multiply by 5 and add 2...
- Ok, 5 times 2.7 is 5*2 + 5*0.7 = 10 + 3.5 = 13.5
- hmm hm hmm hm hmmmm... oh then I have to add 2... so it's 15.5.

- Celestron NexStar 5SE: a five-inch scope, D
_{O}= 125mm- We could round this to 100mm and proceed as above, and we would conclude
that log(D
_{O}) = 2 - But then again you might just notice that this number is 100 × 1.25, and that 1.25 is very close to 1.3
- Since 1.3 has a log of 0.1, then we would say log of our number, 125, is very close to log(100) + log(1.3) = 2 + 0.1 = 2.1.
- We still have to multiply by 5, then add 2, so 5×2.1 = 10.5, then adding 2 we get 10.5 + 2 = 12.5 which is the magnitude limit of a 5 inch (125mm) scope.
- By calculator I get L
_{mag}=12.48, so 12.5 is not bad for doing it in your head.

- We could round this to 100mm and proceed as above, and we would conclude
that log(D
- Meade LX200 10-inch scope, D
_{O}= 254mm- We could round this to 300mm and proceed as above, and we would conclude
that log(D
_{O}) = log(100) + log(3) = 2 + 0.5 = 2.5. - But then again you might just notice that this number is 100 × 2.54, and that 2.54 is very close to 2.5.
- Since 2.5 has a log of 0.4, then we would say log of our number, 254, is very close to log(100) + log(2.5) = 2 + 0.4 = 2.4.
- We still have to multiply by 5, then add 2, so 5×2.4 = 12, then adding 2 we get 12 + 2 = 14 which is the magnitude limit of a 10 inch (254mm) scope.
- By calculator I get L
_{mag}=14.02, so again, an estimate of 14 is not bad for doing it in your head.

- We could round this to 300mm and proceed as above, and we would conclude
that log(D

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Updated 14 November 2012*