I need a brief explanation of logarithms?

Question

Can some bright spark out there give me an easy to understand explanation of logarithms & natural logs, and why would i use them in equations, e.g many times equations are converted to their logarithmic equivalents WHY????

Answer 1

In simple terms logarithms of numbers are ADDED to multiply two numbers together and SUBTRACTED to divide them.

Logarithms in general use are claculated using a BASE of 10

The POWER or index of 10 = logN where N = Number

10^2 = 100 Therefore log 100 =2

10 x 10 x 10 = 10^3 = 1000 Therefore log 1000 = 3

10 x 10 = 10^2 = 100 Therefore log 100 = 2

(10 x 10 x 10) x (10 x 10) = 10^5 = 100000

1000 x 100 = 100000

log 1000 = 3 and log 100 = 2

Adding these log values = 5

i.e. ADDING the powers of 10 gives the log of 100000

Similarly:

(10 x 10 x 10)/ (10 x10) = 10^3/10^2 = 10^1 = 10

log 1000 - log 100 = 3 - 2 = 1 (SUBTRACTING the powers of 10)

N.B.

10/10 =1

Therefore log 10 - log 10 = 1 - 1 = 0 i.e log 1 = 0

From this the value of the log of any number between 0 and 10
to the base 10 lies between 0 and 1

Tables are provided showing these values only.

Any number 10 and above will have a logarithm consisting of a CHARACTERISTIC and a MANTISSA

The CHARACTERISTIC is the WHOLE number part of the log and indicates the decimal point in the final answer.

The MANTISSA is obtained from the log tables.

e.g.What is the log of 300?

The Characteristic will be 2 and the log reference would be checked for 300 in the log tables producing a figure of 0.4771

The log of the number is therefore 2.4771

To find the ACTUAL number the tables of ANTILOGARITHMS must be referred to. N.B. Only the Mantissa is used when checking the Antilog.

In this case Antilog 0.4771 = 3 but, since the Characteristic is 2 the decimal point is moved two to the right giving the result of 300

The question requires a much fuller explanation for Naperian or Natural logarithms which are NOT to the base 10. I would suggest you study the above and then come back with more questions.

Incidentally the Slide Rule is based upon logarithmic scales and multiplictaion and division were based upon two scales SLIDING relative to each other. Adding scale divisions performed mul;tiplictaion and subtracting them performed Division.

Answer 2

The logarithm is the mathematical operation that is the inverse of exponentiation, or raising a number (the base) to a power. The logarithm of a number x in base b is the number n such that b n = x. It is usually written as logb x = n.
Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials. In practical terms, I have found it useful to think of logs in terms of The Relationship:
y = bx
..............is equivalent to...............
(means the exact same thing as)
logb(y) = x
On the left-hand side above is the exponential statement "y = bx". On the right-hand side above, "logb(y) = x" is the equivalent logarithmic statement which is pronounced as "log-base-b of y equals x"; "b" is called "the base of the logarithm", just as b is the base in the exponential expression "bx". And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations. If you can remember this (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms.

(I coined the term "The Relationship" myself. You will not find it in your text, and your teachers and tutors will have no idea what you're talking about if you mention it to them. "The Relationship" is entirely non-standard terminology. Why do I use it anyway? Because it works.)

By the way: If you noticed that the variables are switched between the two boxes displaying "The Relationship", you've got a sharp eye. I did that on purpose, to stress that the point is not the variables themselves, but how they move. Here are some examples of how logarithms work:

Convert "63 = 216" to the equivalent logarithmic expression.
To convert, the base remains the same, but the 3 and the 216 switch sides. This gives me:

log6(216) = 3 Copyright © Elizabeth Stapel 2000-2006 All Rights Reserved

Convert "log4(1024) = 5" to the equivalent exponential expression.
To convert, the base (that is, 4) remains the same, but the 1024 and the 5 switch sides. This gives me:

45 = 1024

Answer 3

So multiplication has division, addition has subtraction and exponents have logarithms.

The reason we have logs is because we need something that undo's powers. Example: 2^x = 9. If I just guess and check we can see the answer needs to be close to 3 because 2^3 = 8 which is close to what we want, but it's not exact. To get the exact answer we made up logs.

What a log equal says is this: Log base 2 of 9 is x, all this is saying is how many 2 are in 9.

But log are very useful in real life too. When you get to calculus you will find out.

Logs, logs, they're big, they're heavy, they're wood.
Logs, logs, they're better than bad, they're good.

Answer 4

I am going to quote from a fantastic discussion about statistical "outliers" in data anaylisis using small (less than 600) sets of data. There are several excellent reference links in these messages you may find helpful. Take a few mins to glance thru them.

Here are 2 related messages to your question, and the link to this discussion thread is provided below that. Interested beginners can find much to proceed with in the first 70 messages and the last several ones. Hope this is a bit of help.

jp's comments:
Warn them about what? The algorithms I've created are based on extensive research... research that I conducted myself first hand... using databases spanning years and several hundred thousand races. The compound numbers generated by my software's algorithms - and I suspect the algorithms of a great many other handicapping software programs on the market today - are hard coded into the program. Individual horses like the one you pointed out won't change the output of those algorithms one single bit - not unless the author/developer decides to go back in and re-code the application and release a new version.

-jp

Traynors definition of algorithm:
Jeff,
Before this goes off the deep end, operationalizing terms may be in order. Specifically, "algorithm." I understand that you hard-wire code that does stuff (a somewhat inelegant way to express it) and that is called an "algorithm." If I "discover" a process that consists of using your application to find the top jockey, on a horse with the highest average purse value, in a turf race that has been switched to the dirt--that is also an algorithm. "Algorithm" simply refers to a step-by-step procedure for solving a problem, or accomplishing something; it includes hard-coded formulas, but also includes a number of other things.

http://www.paceadvantage.com/forum/showthread.php?t=29325&page=1&pp=20

Answer 5

addition is easier than multiplication.this is the principle used in logarithm Napier has painstakingly converted nos 1 to 100 as powers of ten and made what is known as table of logarithms and made another table where the powers of ten are given as their values.so when these nos are to be multiplied you add the powers using the law of indices or exponents using the logarithm tables and use the antilogarithm tables to know the result.instead of expressing as powers of ten if we express as powers of 'e' it is called natural log.
example to multiply 1234 by 5678 we see the logarithms from the tables and add up.3.0713+3.7542=6.8255 and the anti log of .8255 is 6689 and so the product is 8255000
maybe this little explanation helps
the application of logarithms are many.it is too varied to be enumerated/explained here.
if you want i can help you step by step to learn all about logarithms

Answer 6

Read the book.

Answer 7

You should have paid attention in class. I'm not telling you.

Answer 8

i need an explanation too..