What are Logs

Choclolate · 19-01-2011 4:09pm #1

I have heard of common logs and natural logs.

I can use logs and follow all the rules,

But can anyone explain to me what logs actually are?

MathsManiac · 19-01-2011 5:05pm

Try this:
http://www.khanacademy.org/video/introduction-to-logarithms?playlist=Pre-algebra

bnt · 19-01-2011 6:04pm

They can be seen as the "opposite" of Powers. Examples:
- 10³ = 100, so the Base 10 log of 100 = 3
- 2^5 = 32, so the Base 2 log of 32 = 5
- e^12 = 162,754.8, so the Base e log of 162,754.8 = 12.

The Base e log is also called the Natural log, or "LN" on most calculators. e is an irrational number like PI (and just as fundamental), slightly larger than 2.71828. It pops up naturally in certain Calculus operations, e.g. the value of the function e^x is the same as the slope of the same function e^x, something totally unique in mathematics: [latex]\frac{d}{dx}e^x = e^x[/latex].

Fremen · 19-01-2011 11:14pm

[latex]\log_x (Y) [/latex] is shorthand for "the power that you need to raise x to in order to get Y".

Once you have your head around this, you can see for example that
[latex] \log_{3}(Y) [/latex]
is "the power you need to raise three to in order to get Y".

and then
[latex] 3^{\log_{3}(Y)} [/latex] can be read as "three raised to the power you need to raise three to in order to get Y".

In other words,
[latex] 3^{\log_{3}(Y)} = Y [/latex]

Capt'n Midnight · 20-01-2011 12:14am

Most people can add and subtract and if pushed could multiply and divide ( given enough time and care )

For more advanced calculations you can use lookup tables containing LOGS and ANTILOGS (or exponentials as another name)

If you want to multiply two numbers then you can look up their logs in a table, add them together (faster than long multiplication) and then by looking up the antilog you get the result.

So far so good, but suppose you want to find the quadratic root (pay attention one day your life may depend on this one day) of a number ?
look up the log, divide by 4 (easy !) and then look up the antilog and volia !

From this you can see the scales, each division is X times the size of the one before it. Had this graph paper existed in the past things like drying time for wood would have been understood long ago. There was one lad in the Royal Navy who ran such an expiriment for decades but without a way of expressing the results to get a straight line the results didn't have meaning.

There are three numbers used as logs
2 - computers , adding two N bit numbers can be done in 4N clocks , multiplication takes sqr(N) times as long as addition (but only if you use N adders in parallel, otherwise it takes N times as long)
e - good for maths - lots of equations / functions / physical stuff maps well to e
10 - humans can understand, used for decibels etc.

have a look at http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

Human senses like eyes and ears react logitharmically , a sound twice as loud or light twice as bright takes more tha twice as much energy to produce.

decibels are just a way of saying Log to the base 10
3dB = 10^0.3 = twice the power
10dB = 10^1 = ten times the power

FoxT · 20-01-2011 1:46am

Common logs are logs to the base 10.

10^2 = 100, the Common Log of 100 is 2.

Natural logs are logs to the base of e.

Sidebar: What is e?

e = 2.7182818 approx.

e ^ x = 1 + x + (x^2)/2! + ......(x^n)/n!

e is important because when you differentiate e^x you get e^x. Nice!

Back to main point:

Common logs & natural logs are treated in the same way, the difference is that the bases are different.

bnt · 20-01-2011 9:15am

In the days before pocket calculators were widely available, students used log tables to work with large numbers - the kind for which the longhand methods would take forever. For example, if I divide x = 147957 / 326 with a calculator I get 453.86, but if I use base 10 logs I subtract them :
log(147957) = 5.1701
log(326) = 2.5132
log(x) = 5.1707 - 2.5132 = 2.6575
so x = 10^2.6569 = 454.46
You can see straight away that, by rounding the logs to 4 decimal places, I've lost accuracy - but that's exactly what happens when you're looking up the logs from a printed table.

That was another skill you had to learn: the table gave you the logs from 1 to 10, and you had to convert them. For example, you had to express 147957 as 1.47957 x 10^5. Log (1.47957) = 0.1701, then you added the powers of ten, so log(147957) = 0.1701 + 5 = 5.1701. The reverse lookup was, well, that in reverse. And yet some people liked this, and complained when pocket calculators took over. :rolleyes:

Fremen · 21-01-2011 4:24am

Capt'n Midnight wrote: »

Had this graph paper existed in the past things like drying time for wood would have been understood long ago. There was one lad in the Royal Navy who ran such an expiriment for decades but without a way of expressing the results to get a straight line the results didn't have meaning.

Do you have a reference for that? People assume that scientists were very naive back in the day, when in fact they had a huge amount of insight about what they were doing. Given that exponentiation has been around almost as long as arithmetic, I would be very surprised if people were hampered by the lack of logarithms.

Capt'n Midnight · 21-01-2011 12:42pm

Fremen wrote: »

Do you have a reference for that? People assume that scientists were very naive back in the day, when in fact they had a huge amount of insight about what they were doing. Given that exponentiation has been around almost as long as arithmetic, I would be very surprised if people were hampered by the lack of logarithms.

can't remember the name, royal navy dockyards possilby an admiral

while at uni I did a module on cost accounting, the maths was primary school level , I can't even remember if there was any euation solving to get the optimisations. Certainly there was no calculus or numerical analysis which.

Things like economics need maths , the whole economic downturn might have been prevented by game theory simply by showing what would happen if certain assumptions (and they were assumptions) weren't continued to be met.

What are Logs

Comments