LOGARITHMS

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GETTING STARTED

 Let us say ax = b, where b > 0, a> 0 and a ≠ 1.

☛ The exponent x to which base a is raised to obtain a number b is called ‘logarithm of b to the base a

 

Eg. 24 = 16

Hence log 2 16 = 4

☛ The base of the logarithm, a, has to be a positive real number.  Therefore b will also be a positive number.

 

☛ Many a times, a logarithm will be written without a base. Eg. log (125) = log 10(125) = 2.096

It means that the base is 10, and it is called as ‘common log’.

 

☛ Another expression used frequently is ln (x). It denotes natural log where the base is e (Euler’s number) – 2.71828.

Eg. Ln (7.389) = 2

Properties of Logarithms:

(i)log x x =1

Eg. Log 2424 = 1                                             

 

(ii) log a(x. y) = log a x + log ay

Eg. log 232 = log 2(16 x 2)

= log 216 + log 22 = 4+1 = 5

 

(iii) log a(x/y) = log ax – log ay

Eg. log 2(32) = log 2(64/2)

= log 264 – log 22 = 6 – 1 =5

 

(iv) log a1 = 0

Eg. log 101 = 0

 

(v) log a(xp)= p log a x

Eg. log 2(32) = 5 log 2(2)

  = 5 x 1 = 5

 

(vi) log ax = 1/ log xa

Eg. Let us consider log 322

Log 322 = 1/log 232 = 1/5

 

(vii) log ax = log bx/log ba = log x/log a

Eg. Log 464 = log 264/log 24

= 6/2 =3

 

Note: The examples have been considered with integers, all for ease of understanding and calculation. Logarithm of a number can be a decimal as well.

PRACTICE PROBLEMS

0

Logarithms

1 / 21

Simplify :

log61296 + log2128 + 3log525.

2 / 21

Find the relation between x and y, if

3 log25x = 4 log5y + 5.

3 / 21

If (log2438) (logx3) = 1,

 

find x.

4 / 21

Find the value of

 

log 710(2401)5

5 / 21

N is a 40 digit number. Between what values would log10N lie?

6 / 21

If log 5 243 = 2.385,

then the value of log527 is:

7 / 21

Find the value of log 8(32)7

8 / 21

FInd the value of x if

log (4 x+4)+log (x+1)-log (10 x + 2) = log 2

9 / 21

Find the value of x if

log√5 (x + 584) = 8

10 / 21

If log 2 = 0.3010 and log 3 = 0.4771, the approximate value of log5 729 is:

11 / 21

Find number of zeros after decimal point in

 

(2/3)400

 

if log 2 = 0.301
log 3= 0.4771

12 / 21

Find the value range of A if

A = log 860

13 / 21

If 5y = (0.01)x = 500, what is the value of (1/y - 1/x)?

15 / 21

Find the value of x when log2(5x+3)=log3(10x-1)+1

16 / 21

Find the number of digits in

 

25230

given that log 6 = 0.778

log 7 = 0.845

18 / 21

The value of log2log3log3log3279 is:

19 / 21

If value of log105 = 0.699, find approx. value of

log510

20 / 21

If log10 2 = 0.3010, and log10 3 = 0.4771, the value of log10.40.5 is:

21 / 21

Find the value of log10600

if
log102=0.301
log103=0.477

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