Logarithms

In general Terms, Logarithm refers to the number that shows how many times a base number (such as ten) is multiplied by itself to produce a third number (such as 100).

Logarithm was developed by John Napier and is very useful tools for mathematical computation.

If n^x= a, the logarithm of a, with n as the base, is x. It is symbolically expressed as log_n a = x.

The equations, a^x=n and x=log_an are only transformation of each other, explaining the relationship between the 3 components, base, index & logarithm.

Logarithm may be defined as the number, which, to a given base as the index (or the power) to which the Base should be raised to yield the given number.

In the expression, 2⁴ = 16, 2 is termed to as the base, 4 as the index (or power), 16 is the number. This could also be  expressed as log₂16 = 4 and is read as the Logarithm of 16 to base 2 is 4.

In the expression 5^-3 = 1/5³ =1/125, So, log₅ (1/125) = log ₅(1/5³) = log₅(5^-3). So, log₅ (5^-3) = -3

Logarithm Rules

• The logarithm of 1 on any base is 0.Ex. Log₁₀ 1=0, Log ₂ 1=0
• The logarithm of a number  with same base is 1. Ex. log₁₃13 = 1.
• The logarithm of a product to base a is equal to sum of the logarithms of the individual numbers which constitute the product to the same base a.

log_n(AxB) = log_nA + log_nB. Ex. Log_n (15) = Log _n (5X3) = Log _n 5 + Log_n 3

• The logarithm of a fraction to the base a will be equal to the difference of the logarithm

of the numerator to the base a, and the logarithm of the denominator to base n. 

log_n (A/B) = log_nA − log_nB. Ex. Log _n 25 = Log _n (100/4) = Log _n 100 – Log _n 4

• The logarithm of a number raised to any power, integral or fractional, is equal to product of that number and the logarithm of the number raised to base a.

log_n (M^p) = p.log_n M. Ex Log_n 5² = 2 Log_n 5

Natural Base : 10 is considered as natural base (common logarithm). If no Base is mentioned, it is assumed that the Base is 10. So, log 234 indicates log ₁₀ 234

Logarithm Rule – Problem

Rules of Logarithm – Problems & Solutions

Ex. Solve 7 log (32/15) + 5 log (25/24) + 3 log (81/80).

Step 1= 7 (log 32 – log 15) + 5 (log 25 – log 24) + 3 (log 81 – log 80)

Step 2= [7 {log 2⁵ – log (3 x 5)}] + [5 {log 5² – log (2³x3)}] + [3 {log 3⁴ –log (2⁴x5)}]

Step 3= {7{5 log 2– (log 3 + log 5)}] + [5 {2log 5 – (3log 2 +log 3)}] + [3{4log 3 – (4 log 2 + log 5)}]

Step 4= [7 (5 log 2 – log 3 – log 5)] + [5 (2 log 5 – 3 log 2 – log 3)] + [3(4 log 3 – 4 log 2 – log 5)]

Step 5= 35 log 2 – 7 log 3 – 7 log 5 + 10 log 5 – 15 log 2 – 5 log 3 + 12 log 3 – 12 log 2 – 3 log 5

Step 6= 35 log 2 – 15 log 2 – 12 log 2 – 7 log 3 – 5 log 3 + 12 log 3 – 7 log 5 + 10 log 5 – 3 log5

Step 7= 35 log 2 – 27 log 2 – 12 log 3 + 12 log 3 – 10 log 5 + 10 log 5

Step 8= (35 – 27) log 2 + (−12 + 12) log 3 + (− 10 + 10) log 5

Step 9= 8 log 2 + 0 log 3 + 0 log 5

Step 10= 8 log 2 + 0 + 0

Step 11= 8 log 2.

Logarithms Rule – Problems

= log₂ (25 / 100)=log₂ (1/ 4) = log₂ (1/ 2²)=  log₂ (2^-2) = -2

log [(abc²) / d³] = (log a + log b + log c²) – log (d³) = log a + log b + 2log c – 3log d

log[(x√5)/z⁵] = logx + log (√5) –  log (z⁵) = log x + log (5^1/2 ) – 5 log z= log x +½(log5 )- 5 log z

Change of Logarithm Base

Often, you need to change logarithm base to get the logarithm of a number from one Base to another base.

The relation between the logarithms of a number to different base

Log_a m = log _bm x log _ab

logarithm of a number (m), with existing base(a), i.e Log_a m = Logarithm of the number (m) with  new Base (b) x Logarithm of the new base (b), with respect to the existing base (a)

Ex. Change the base log₂ 45 to common logarithmic base.

Log_am = log_bm / Log_ba, So, log₂45 =  Log₁₀45 / Log₁₀2 = log₁₀45 – log₁₀2

Properties of Logarithm

Properties of Logarithm

• log_amn =log_am + log_an. Logarithm of product of two number is equal to sum of the logarithm of each number.
• log_a (m/n) =log_am – log_an. Logarithm of fraction of two number is equal to difference between the Logarithm of numerator and Logarithm of Denominator
• log_amⁿ =n(log_am) : Logarithm of power (exponential) of number is equal to product of the power  (exponential) and the Logarithm of the number
• log_aa = 1 : Logarithm of a number with same base is always 1
• log_a1 = 0 : Logarithm of 1 is always zero (irrespective of value of Base), because a⁰=1, for any value of a.
• log_am = 1/ (log_ma). So, log_am X  log_ma = 1. When Base and the Number is interchanged, the logarithm value is inversed, and the value of their product is always 1
• log_bn = ( log_an) / (log_ab) : When Base is changed, the logarithm value of the number with new base is the ratio between the logarithm value of the number with current base and the logarithm value of the Base with Base of the number
• log_ab x log_bc x log_ca =1 : The product of logarithm of 3 numbers in series with their Base is 1
• log_amⁿ = (n/b) log_bm : The logarithm of a number raised to a power (exponent) is equal to the product of the number divided by the new Base and logarithm of the number with new Base

Properties of Logarithm – Problems

Properties of Logarithm – Problems & Solutions

Ex. Find the value of log₂32 – log₈2

Step 1 : log₂(2 ⁵ )  – [1/ (log₂8)]

Step 2 : log₂ 2⁵ – [1/ log₂ (2³)]

Step 3 : 5 (log₂ 2) – {1/ (3 log₂ 2)  =  (5×1) -1/(3×1)= 5-1/3= 14/3

So, the value of the expression is 14/3

Logarithm & Antilogarithm Tables 

Logarithm Table ( Log Table) & Antilogarithm Tables (Anti Log Table) are were wisely used in performing large calculations (of multiplication, division, squares, and roots) without using a calculator.

Log Table & Antilog Tables are printed in Paper, from which the value of Logarithm and then the reverse Antilogarithm is found. However, with advent of electronic calculator, now more complex arithmetic an scientific  calculations can be performed using such digital calculators. So, Log Tbales are outdated and no more used.

However, a brief commentary and Illustration on Log Table & Antilog Tables are included to comply syllabus still included in some courses.

Logarithm & Antilogarithm Tables – Problems

Logarithm & Antilogarithm Tables – Problems and Solutions

Ex.Find the value of  ∜0.06754.

Let x =∜0.06754= (0.06754)^1/4

Taking log on both sides, we get

log x = (1/4) x log(0.06754)

log x = (1/4) x (–2 + .8296)

log x = (1/4) x {–2 + (-2+2)+ .8296}

log x = (1/4) x [{–2 + (-2)}+ (2)+ .8296)}

log x = (1/4) x (-4+ 2.8296)

log x = [{(1/4) x (-4)} + {(1/4)x2.8296)}]

log x = -1 + .7074

So, x = antilog (-1 + .7074) = 0.5098

Properties of Logarithm – Problems

Logarithm – Problems and Solutions

Ex. If log_abc = x, log_bca = y, log_cab = z, prove that

{(1/(x+1)} + {(1/(y+1)} + {(1/(z+1)} + 4=5

Step 1 : We may write

x+1 = log_abc + 1 = log_abc +  log_aa = log_aabc

y+1 = log_bca + 1 = log_bca + log_bb = log_babc

z+1 = log_cab + 1 =  log_cab + log_cc = log_cabc

Step 2 : So, the LHS of the above expression  may be written as

LHS= {(1/ log_aabc) + (1/ log_babc) + (1/ log_cabc)} +4

LHS = (log_abca + log_abcb + log_abcc) +4

LHS = log_abcabc +4  = 1+ 4 = 5

Characteristics and Mantissa in Log Tables

Mantissa in Log Tables

Common logarithms : Logarithms calculated to the base 10 (Natural Base) are called common logarithms. Common logarithm has two parts, the whole part and the decimal part.

• Characteristic : The whole part of the logarithm of a number is called Characteristic
• Mantissa.: The non-negative decimal part is called the Mantissa.

Example : When value of logarithm is 2.4567. The value of Characteristic is 2 and value of Mantissa is .4567

Determination of Characteristics

• Characteristics of Number greater than 1 : When the logarithm of a number is greater than 1, first determine number of digit before decimal point. Then deduct 1 from number of digits before decimal point to get the get the characteristics.

Examples : Characteristics of 2.454.  2.454> 1. number of digit before the decimal point = 1. So, characteristic of log 2.454= 1 – 1 =0

Characteristics of 245.8. 245.8> 1. The number of digit before the decimal point = 3. Hence, characteristic of log 245.8= 3 – 1 =2

• Characteristics of Number less than 1 : Logarithm of a number less than 1 is negative and the characteristic value is one more than the number of zeroes immediately before the decimal point.

Examples : Characteristics of .2454 The number .2454<1. Number of zeroes before the decimal point = 0. So, characteristic of log .2454= – (0 +1) = -1 (or`1 )

Characteristics of.002454 . The number .002454 <1. The number of zeroes before the decimal point = 2. So, characteristic of log .002454 = – (2 +1) = -3 (or `3 )

Computation of Mantissa

The Mantissa is same for the same figures in the same order and does not depend on the position of the decimal point.

Only significant figure is considered and the Mantissa of the logarithm of a number is determined by using logarithm table. 

Ex. Mantissa of 2454, 2.454, .2454, .0002454, is same and its value can be determined from the log table.

Finding value of  Mantissa from Log Table.

The main body of the Logarithmic  Table (referred as Log Table) gives the mantissa of the logarithm of numbers of 3 digits or less, while the mean difference Table provides the increment for the fourth digit.

Step 1 : Ignore the decimal point (if any) of the given number.

Step 2 : In the first column (from left) in the table, locate the first two digits of the given  number. If however, the given number consists only one digit, we make it 2 digited number by adding a    zero its right.

Step 3 : Along the row corresponding to these two digits, locate the entry under the column  headed by the third digit in the given number.

Step 4 : For the fourth digit, move further along that row and suitably locate the entry in the column (of mean difference) header of the fourth digit. Now, add this entry to the number obtained in step 3.

Step 5 : The final entry is thus obtained from the table, preceded by a decimal point is the required mantissa of the given number.

Determination of Logarithm – Problems

Computation of Logarithm using Log Tables

Find the value of log 24.57.

Determination of characteristics : 24.57>1. Number of digits before decimal place is 2. So, the Characteristics is 2-1 = 1

Determination of Mantissa from Log Table

Step 1 : Ignore the decimal point first.

Step 2 : Locate24(the first 2 digit ) in the first column.

Step 3 : Move along the row of 24 and locate the entry in the column head 5 (third digit). This entry is 3892.

Step 4 : Move further along the same row and locate the entry in column of Mean Difference under column head 7 (the fourth digit). This entry is 12.

Step 5 : Add 3892 and 12 to get the value of Mantissa, i.e 3904.

Step 6: Write the characteristics (i.e 1). Put decimal point and write the mantissa. So the value of log 24.57 = 1.3904.

Antilogarithms

If x is the logarithm of a given number n with a given base, then n is called the antilogarithm (antilog) of x to that base.

If log(n) = x, then n = antilog x

If log 2457 = 3.3904   then Antilog 3.3904 = 2457

Ex. Find the value of log 50 if log 2 = .3010                                   

Log 50 = log (100/2) = Log 100 –  log 2 = 2-.3010 = 1.6990

Determination of Antilog value from Antilog Table

– Step 1 : Ignore the characteristic of the number (Logarithmic Value) for which Antilog is to computed. Make sure that decimal part is positive.

– Step 2 : Locate the value in the antilog table for the value (the process is similar to finding from Log Table). 

– Step 3 : Insert the decimal point in the value obtained from Anti Log Table. The number of digits before decimal point will be 1 more than the characteristic of Logarithmic value (the number for which the antilog value is being computed)

Ex. Find Antilog 1.3246.

From the antilog table, for mantissa .324, the number = 2109. For mean difference 6, the number = 3. So, for mantissa .3246, the number = 2112

The characteristic is 1, therefore, the number must have 2 digits in the integral part.

Hence, Antilog 1.3246 = 21.12

Ex. Find Antilog (– 2.3246).                                                              

−2.3246 = −3 + (3 − 2.3246) = (−3) + .6754. Now find Antilog for  .6754. From Antilog Table, you get 7332 for .675 and then the mean difference for 4 is 4. So, adding together, you get Antilog (.6754) = 7332+4=7336. The  characteristic is (-3).

So, the number is less than 1 and there must be 2 zeros before the decimal place. So, the value would be 0.007336. So, Antilog (−2.3246) = 0.007336

Computation of Logarithm and Antilogarithm

Computation using Logarithm and Antilogarithm Tables

Ex. Find the value of (391.1 x (0.08354)^1/5 / (3.843)³

y = [{(391.1 x (0.08354) ^1/5} / (3.843)³]

Taking log on both sides, we get

Log y  = log 391.1 + [(1/5) x (log 0.8354)] – {3 (log 3.843)}

LHS = 2.5923 +  [(1/5) {(-1 + 0.9219)}] – [3 x (.5847)]

LHS = 2.5923 +  [(1/5) {(-5 + 4.9219)] – 1.7541

LHS = 2.5923 +  [(-1 + 0.09844) – 1.7541]

LHS = 2.5923 +  (-1) + 0.09844 – 1.7541

LHS = 2.5923 -1 + 0.09844 – 1.7541 = 0.8226

So, y= antilog (0.8226) = 6.646

Click Here to play Hindi Video of student using example of Logarithms

To see PDF