LOGARITHM

Logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number. That is, to make it equal to the given number.

Let us consider the three quantities, sat a, x and n and they can be related as follows. 

If a^x  =  n, where n > 0, a > 0 and a ≠ 1, then x is said to be the logarithm of the number n to the base 'a',  symbolically it can be expressed as follows. 

log_an  =  x

Here, 'n' is called as argument.  

Fundamental Laws of Logarithm

Law 1 : 

Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base. 

That is, 

log_amn  =  log_am + log_an

Law 2 : 

Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base. 

That is, 

log_a(m/n)  =  log_am - log_an

Law 3 : 

Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.  

That is, 

log_amⁿ  =  nlog_am

Relationship between Indices and Logarithms

Example 1 : 

If log_am  =  x, then

m  =  a^x

Example 2 : 

If k  =  x^y, then

log_xk  =  y

Multiplication of Logarithms

Multiplication of two or more logarithms can be simplified, if one of the following two conditions is met. 

Condition (i) : 

The argument of the first logarithm and the base of the second logarithm must be same. 

Condition (ii) : 

The base of the first logarithm and the argument of the second logarithm must be same.

Let us see how the multiplication of two logarithms can be simplified in the following examples. 

Example 1 :

Simplify : log_ab ⋅ log_bc

In the above two logarithms, the argument of the first logarithm and the base of the second logarithm are same. 

So, we can simplify the multiplication of above two logarithms by removing the part circled in red color. 

Example 2 :

Simplify : log_xy ⋅ log_zx

In the above two logarithms, the base of the first logarithm and the argument of the second logarithm are same. 

So, we can simplify the multiplication of above two logarithms by removing the part marked in red color. 

Change of Base

If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation. 

log_ba  =  log_xa ⋅ log_bx

log_ba  =  log_xa / log_xb

More clearly, 

In the above example, the base of the given logarithm 'b' is changed to 'x'. 

In this way, we can change the base of the given logarithm to any other base. 

Inverse Properties of Logarithms

Some other Properties of Logarithms

Property 1 : 

If you switch a logarithm from numerator to denominator or denominator to numerator, we have to interchange the argument and base. 

Examples : 

log_ab  =  1 / log_ba

5 / (2log_xy)  =  5log_yx / 2

Property 2 :

Logarithm of any number to the same base is equal to 1. 

Examples : 

log_xx  =  1

log₃3  =  1

The above property will not work, if both the base and argument are 1.

That is, 

log₁1  ≠  1

The reason for the above result is explained in the next property. 

Property 3 :

Logarithm of 1 to any base is equal to 0. 

Examples : 

log_x1  =  0

log₃1  =  0

log₁1  =  0

Notes

1.  If the base is not given in a logarithm, it can be taken as 10. 

Thus, 

log10  =  log₁₀10  =  1

log1  =  log₁₀1  =  0

2.  Logarithm using base 10 is called Common logarithm and logarithm using base 'e' is called Natural logarithm. 

Here, e  ≈  2.718 called exponential number.   

Practice Problems

Problem 1 :

Find the logarithm of 64 to the base 2√2.

Solution : 

Write 64 as in terms of 2√2. 

64  =  2⁶

64  =  2⁴⁺²

64  =  2⁴ ⋅ 2²

64  =  2⁴ ⋅ [(√2)²]²

64  =  2⁴ ⋅ (√2)⁴

64  =  (2√2)⁴

Then, 

log_2√264  =  log_2√2(2√2)⁴

log_2√264  =  4log_2√2(2√2)

log_2√264  =  4(1)

log_2√264  =  4

Problem 2 :

If log_abc  =  x, log_bca  =  y and log_cab  =  z, then find the value of

1/(x+1) + 1/(y+1) + 1/(z+1)

Solution : 

x + 1  =  log_abc + log_aa  =  log_aabc

y + 1  =  log_bca + log_bc  =  log_babc

z + 1  =  log_cab + log_cc  =  log_cabc

1/(x+1)  =  1 / log_aabc  =  log_abca

1/(y+1)  =  1/log_babc  =  log_abcb

1/(z+1)  =  1/log_cabc  =  log_abcc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  log_abca + log_abcb + log_abcc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  log_abcabc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  1

Problem 3 :

If a = log₂₄12, b = ₂₄36 and c = log₄₈36, then find the value of

1 + abc

Solution : 

1 + abc  =  1 + log₂₄12 ⋅ log₃₆24 ⋅ log₄₈36

1 + abc  =  1 + log₃₆12 ⋅ log₄₈36

1 + abc  =  1 + log₄₈12

1 + abc  =  log₄₈48 + log₄₈12

1 + abc  =  log₄₈(48 ⋅ 12)

1 + abc  =  log₄₈(2 ⋅ 12)²

1 + abc  =  2log₄₈24

1 + abc  =  2log₃₆24 ⋅ log₄₈36

1 + abc  =  2bc

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6