Math Is Fun Forum

wasee

Member

Registered: 2007-08-12

Posts: 2

Logarithm Tables Calculation

Can someone tell me how the logarithm tables were found? Well, I know they are simple for numbers which are multiples of 10 but what about numbers like 2,5 ,0.01 etc.? For example, we know that log 10 = 1 and log 100 = 2, it is simple because we know that 10 ^1 = 10 and 10^2 = 100  but how do we calculate 10^x = 2 ,in other words log 2 = x ( I don t know the answer). The problem is that 2 is not an integral mutiple of 10 so how were these tables calculated ?

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Logarithm Tables Calculation

for more complicated functions, simple formulas like f(x) = x^2 - 2x + 3 will no longer suffice. There are some functions which simply cannot be expressed by a finite number of elementary operations (like addition, subtraction, multiplication, etc) and so these are usuallly called non elementary functions. However, if you begin to consider functions that consists of infinitly many operations, that opens the doors to non elementary functions and allows us to evaluate and analyze them.

Take for example the power function f(x) = e^x, you can show that
e^x = 1 + x + x^2 + x^3/3! + x^4/4! + x^5/5!...and so on and so forth

you can see that a pattern exists here. But how can we hope to evaluate a function with infinitly many terms? well we can't, however, we can get an approximation of the function by using a finite number of terms. This is because the terms are getting smaller and smaller as you move down the line, and the additional terms make less and less of a difference. In fact, there exists methods which could, for instance, tell you that if you use the first 10 tems and add them, your answer is correct to up to 5 decimal places. So really, how many terms you use depends on how accurate your answer needs to be.

Likewise, for something like log(x) a series can be developed to calculate the value to any given level of accuracy.

the concept of expressing a function as the sum of an infinite series is discussed in second semester calculus. You'll learn all about it.

Its also worth mentioning that there is usually more than one way to skin a cat, and more than one way to calculate a function. For instance, you could calculate the sine of an angle with a compass and a ruler! or measure volume by placing an object in a bowl of water and measuring how much it displaces, or find an object's center of gravity by hanging it from several points with a string. As for methods that aren't infinite series' that could calculate ln(x), the function f(x) = n* (x^(1/n) - 1)  is very close to f(x) = ln(x) where n is some very big number, like 1 million, and it gets closer and closer the bigger n is.

Another method, is to try to calculate the area under the curve y = 1/x between x = 1 and u, this is in fact equal to ln(u). There is another method called newtons method which might be useful in determining its value.

Anyway, i'm just babbling now but as you can see, there are lots of ways to do it and you'll learn about pretty much all of the ones i mentioned in second semester calculus, if you ever take it. So put your mind at rest!

Last edited by mikau (2007-08-16 20:01:48)

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A logarithm is just a misspelled algorithm.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Logarithm Tables Calculation

http://en.wikipedia.org/wiki/Logarithm# … _logarithm

Some infinite series that converge to natural logarithms. To convert them to base 10 logs, simply divide by the natural log of 10.