How do you do calculations using log tables?

 Let us see how we proceed with the following calculation using log and anti-log tables.

Look at the calculation below:

 

So, first let us convert all the numbers in the calculation up to a maximum of 4 significant figures, with only a single digit before the decimal point, i.e., it will be as below:

Let us say the value of the calculation is = n.

That implies,  

Now, we need to extract the n value by taking the anti-logarithm. 

So, the value of the given calculation is = 8.563 x 10^1 = 85.63.

How do I use anti-log tables?

To determine the anti-log value of a given number, let's use 5.4655 as an example. Antilog(5.4655) =?

The first step is to determine the ‘characteristic’ value and mantissa.

For the given number, the characteristic value is 5, and the mantissa is .4655. In the mantissa, look at the first two digits (.46), go to the anti-log table, and look at the .46 row. Note the value from the row, under column 5. (= 2917); also note the mean difference value, under column 5. (= 3). Now add both of them (2917 + 5 = 2922).

Now, let's discuss the placement of the decimal point. By default, we place the decimal point after the first digit. Next, we multiply the value by 10^(the characteristic value).

i.e., in our example, = 2.922 x 10^(5). Therefore, anti-log(5.4655) = 2.922 x 10^(5).

Let's examine a few additional examples to enhance our understanding.

	Characteristic value	Look-up Antilog table	Mean difference	Total value
Anti-log(8.579) =?	= 8; Mantissa = .579	Look up in the .57 row, under col. 9. (= 3793).	Not needed here, since there is no 4th digit in the mantissa.	Place the decimal point => 3.793. Decide the 10 power => 10^(8). Total = 3.793 x 10^(8)
Anti-log(-3.7586) =? First, make the characteristic positive. = -3.7586 + 4 – 4 = -4 + 0.2414.	= -4; Mantissa = .2414	Look up in the .24 row, under col. 1. (= 1742).	Under col. 4 in mean difference (= 2).	= 1742 + 2 = 1744. Place the decimal point => 1.744. Decide the 10 power => 10^(-4) Total = 1.744 x 10^(-4)

 

 

How to use Log tables ?

Let us take one example; let us find the log(345.687) value. For most of the numerical calculations, the base will be 10. 

We need to express the given number with only one digit before the decimal point, see below.

345.687=3.456×10^(2)

Now, the power of 10, i.e., 2 is our characteristic value, for the given number. Now, we need to look up the log table with the four digits 3456. Take the first two digits i.e., search for 34 in the log table, in that row, note the number in the column corresponding to 5, (= 5378). Now, in the same row, look at the mean difference value under column 6, (= 8). Add these two, i.e., 5378 + 8 = 5386. 

Now, place the characteristic value before this, i.e., 25386, and place the decimal point after one digit, i.e., = 2.5386. This is the value of log(345.687) =2.5386.

Let us look at a couple more examples below, which makes it clearer.

	Characteristic value	look-up Log-table	Mean difference	Total value
log(4567.453) = ?	= 4.567×10^3 → 3	45th row, under column 6 = 6590	Same row, under col 7 in mean diff. = 7.	6590 + 7 = 6597. Place characteristic value => 36597.  Place the decimal point => 3.6597
log(0.9668) = ? = log( 9.668 x 10^(-1) ) = log(9.668) -1.	= 9.668×10^0 → 0	96th row, under column 6 = 9850	Same row, under col 8 in mean diff. = 4.	9850 + 4 = 9854. Place characteristic value => 09854. Place the decimal point => 0.9854. Now, the final value = 0.9854 – 1 =   = - 0.0146. (yes, that is negative !)

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How can we prove Cauchy-Goursat theorem ?

Cauchy - Goursat theorem says, the integral of an analytic function  f(z) over a closed contour C ,  is zero. Mathematically,

 

( Function has to be analytic at all the interior points and on the contour C as well.)

To prove this, let's start with an analytic function f(z). Consider

 where C is any simple closed contour in the Z-plane. By knowing  
f(z) = u(x,y) + i v(x,y), and dz = dx + i dy , the above integral becomes 

 

where vectors V_1 = (u, -v) , V_2 = (v, u) and dr = (dx, dy).
Now, the two closed loop integrals on the right hand side can be converted into surface integrals, by using Stokes theorem. Hence, we can write,

 

 We can realize that these two integrands goes to zero, since the function  f(z), is an analytic function. Hence it satisfies the Cauchy-Riemann conditions,

So, for an analytic function, it is guaranteed that,

 

which implies

 

Hence the theorem is proved. 

What does it mean by a complex function f(z) ?

In complex calculus, we define the complex number  z as,

z = (x+i y),  where x and y are real numbers and  i = sqrt(-1)

The plane formed by x & y  is called as the ' z-plane ' .  The function  f(z)  is again a complex number, as  z is complex and we define the function  f(z) as follows.

f(z) = w = u + i v , where u & v are real and depend upon the value of z. Hence they are functions of x & y.

i.e., u = u(x, y) , v = v(x, y) ;  and the plane formed by u and v is called as the w-plane. So, the function  f(z) can be viewed as a mapping from the z-plane to a w-plane, for a specified mapping rule (for  a defined function mathematically). 

Lets see some examples to make it clear how the function f(z) maps certain geometries in z-plane into a w-plane under certain specified mapping rule.

Ex. 1.  Say we've y = mx (a straight line passing through origin ) in the z-plane, lets' see how this geometry maps into a w-plane, under the mapping rule  f(z) = z^2.

we've f(z) = z^2 = (x + i y)^2 = (x^2 - y^2) + i (2xy)

that gives  u(x, y) = (x^2 - y^2)  and  v(x, y) = 2xy. 

But we've the relation y = mx  in z-plane, so, by substituting this relation into u & v,  we get,

u = (1-m^2)  x^2 ,  

v = 2m  x^2

by eliminating x from the above two eqns, we can obtain the relation b/w  u & v. as,

v = [2m / (1-m^2)] u , which is also a straight line with slope 2m / (1-m^2).

To represent it  pictorially,

Similarly, one can think of any arbitrary mapping rule and try to visualize how does it maps the given geometry in the z-plane into a w-plane. For example, if you take mapping rule to be exp(z),  then verify yourself that, it maps into a spiral in the w-plane.
If its not possible to obtain the relation directly  b/w  u and v  by the elimination of variable x, then one can go for either mathematica or any numerical technique to see the parametric plot  b/w  u & v.

That tells you the geometry in w-plane.