Using Logarithmic Tables

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• Anonymous
• Narayana Rao KVSS

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Last edited: Apr 7, 2012

Exported: Apr 30, 2012

Original URL: http://knol.google.com/k/-/-/1zb6eis38d7or/15

Express the given number "n" in the form of m * 10^p       where 1≤m<10 and p is an integer(positive or negative whole number).

 

For example number 2 is expressed as 2*10⁰

 

log n become equal to p + log m

 

log 2 becomes equal to 0 + log 2

 

p is called the characeristic and log m is called the mantissa. Mantissa is read from the logarithmic tables.

 

Logarithmic tables are show three sets of columns

i) the first set of column on the extreme left contains numbers from 10 to 99.

ii) in the seocnd set there 10 columns headed by 0,1,2,...,9

iii) after this, in the third set there 9 more columns headed by 1,2,3...9. These are known as mean differences.

 

As 1≤m<10, the mantissa is for a number between 1 and 10. Hence the interpretation of the first set of column in the table is 1.0 to 9.9, If you add the digit in the second set one more digit is added to the number. Which mean 1.0 becomes 1.01. If we add a digit in the third column on more digit is added to the number. Which means 1.01 becomes 1.011.

 

Hence log 2 = 0 + 0.3010 = 0.3010

 

How to see its antilogarithm.

 

Antilogaritm tables are written from .00. If mantissa of a logarithm is .00, then antilogarithm is 1.000

Antilogarithm of .3010 is equal to 2.000

As the characteristic of the number is 0 the number is 2.0*10⁰. Which is equal to 2.

 

Suppose the problem is to find 2^(1/6). It is 2 to the power (1/6).

 

When we take logarithms, it becomes (1/6)* log 2 which is equal to (1/6)*(0.3010) = 0.0617 (rounded)

 

What is antilogarithm of 0.0617 = 1.153*10⁰. = 1.153

 

So the answer of 2^(1/6) is equal to 1.153.

Logarithm - Definition

If a^x = y then log_ay = x

log is the abbreviation of the word logarithm

Fundamental Laws of Logarithms

log MN = log M + log N

log M/N = log M - log N

log Mⁿ = n log M

log 1 = 0

log_aa = 1 (log of any positive quantity of the same base is always one.

Systems of Logarithms

Common logarithms to the base 10

Natural logarithms to the base e

Standard Form of Decimal

If k is a positive number we can express it as

k = m*10^p

Where m is a decimal number such that 1≤m≤10 and p is an integer.

This form m*10^p is called standard form of decimal.

Finding Log N from tables

Find the characteristic of N.
Find the mantisssa
Log N = Charateristic + Mantissa

Antilogarithms

If log n = m

n = antilog m

Logarithms - Basic points

Let a,b be two positive real numbers and a≠1. The real number x such that a^x = b is called logarithm of b to the base a.

X = log_ab

Theorems

1. If a is a positive number and a≠1 then log_aa = 1.

2. If a is a positive number and a≠1 then log_a1 = 0

3. If a,m are positive real numbers, a≠1 then a to the power log_am = m.

4. If a,m,n are positive real numbers and a≠1 then log_amn = log_am + log_an

5. If a,m,n are positive real numbers and a≠1 then log_am/n = log_am - log_an

6. If a,m,n are positive real numbers and a≠1 then log_amⁿ = nlog_am

7. If a,b,n are positive real numbers and a≠1, b≠1 then log_am = log_bm* log_ab

8. If a>1, then x>y => log_ax > log_ay

9. If 0y => log_ax < log_ay




















e = 1+1/1! +1/2! + 1/3! + ¼! + …∞

e = lim n →∞ (1+(1/n)) ⁿ


If aⁿ = x, the log_ax = n.

Log (1=x) when |x|<1 = x-x²/2 +x³/3 - x⁴/4+…∞

Logarithms-1

Given a positive real number y, there is one and only one real number x such that a^x = y, we call this number x as logarithm of y to the base a.

Logarithm gives the power to which the base is to raised to get the number.

Logarithms can be defined for any base.Logarithms defined to the base 10 and to the base e are tabulated and used more widely. Logarithms to the base 10 are called common logarithms and logarithms to the base e are called natural logarithms.

Many books follow the convention of writing natural logarithms as ln y and common logarithms as log y.

In mathematics books as logarithms are given for various bases, the base may be specified.

Some properties

1. log_a1 = 0, a>0, and a ≠0
2. log_aa = 1, a>0, and a ≠0

Good online material

http://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx