 Common Logarithm and Natural Logarithm:

• Common logarithm has base 10 ($b = 10$) and is denoted as $\log(x)$, while natural logarithm has base $e$ (Euler’s number) and is denoted as $\ln(x)$.

### Logarithm Rules:

• Product Rule:  $log_{b}\left ( xy \right )= log_{b}\left ( x \right )+log_{b}\left ( y \right )$
• Quotient Rule: $log_{b}\left ( \frac{x}{y} \right )=log_{b}\left ( x \right )-log_{b}\left ( y \right )$
• Power Rule: $log_{b}\left ( x \right )^{n}=nlog_{b}\left ( x \right )$

A logarithm denote as the contradictory of power. In other terms, if we go for a logarithm of a particular value, we unknot exponentiation. Logarithm Questions and Answers are discussed below:

For instance: If the base is taken as b = 3 and increase it to the power of k = 2 we get the result as 32. The result is referred as c, showed by 32 = C. The rules of exponentiation can be used to evaluate that the result is C =32 = 8.

Given an example, consider that someone inquired “2, raised to which power is equivalent to 16”? The result will be 4. It is further articulated by the logarithmic calculation, i.e. log2 (16) = 4, which is further spoken as “log base two of sixteen is four.”

Logarithm form

• Log2(8) = 3
• Log4(64) = 3
• Log5(25) = 2

Exponential form

• 2^3= 8
• 4^3= 64
• 5^2= 25

Generalizing the examples above leads us to the formal definition of a logarithm.
Logb (a) =c ↔ bc =a

Both the equations define the similar link where:
‘b’ is considered as the base,
c is considered as the exponent
a is considered as the argument

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1. If the value of log 3 = 0.477, then find the number of digits in 336 18

18

47.06%

3

3

35.29%

20

20

11.76%

24

24

5.88%

Number of digits in 336 ≈ ⌊log10(336)⌋ + 1
Number of digits in 336 ≈ ⌊log(112 * 3)⌋ + 1
Number of digits in 336 ≈ ⌊(log(112) + log(3))⌋ + 1
Number of digits in 336 ≈ ⌊(2.049 + 0.477)⌋ + 1
Number of digits in 336 ≈ ⌊2.526⌋ + 1
Number of digits in 336 ≈ 2 + 1
Number of digits in 336 ≈ 3

So, the number 336 has 3 digits. 2. If logx (5/18) = 1/2 , then find the value of x: 456/12

456/12

12.5%

25/324

25/324

68.75%

6.25%

566/18

566/18

12.5%

Solution: logx5/18 = 1/2

x1/2 = 5/18

√x = 5/18

Squaring both sides

x = (5/18)2

x = 25/324 3. The value of log3 27 is : 3

3

72.22%

2

2

16.67%

7

7

5.56%

8

8

5.56%

Explanation : log3 33

= 3 4. If log 5 = 0.698, find the number of digits in 525 3

3

23.53%

18

18

47.06%

4

4

11.76%

6

6

17.65%

Solution: log (525) = 25* log 5

= 25* 0.698

= 17.45

= 18 ( approx.) 5. Solve : px = qy z/x

z/x

11.76%

y/x

y/x

70.59%

X/z

X/z

11.76%

A/x

A/x

5.88%

Solution: log px = log qy

x log p = y log q

log p / log q = y/x 6. Solve the given logarithmic equation:

log7x = 3 324

324

13.33%

343

343

73.33%

289

289

6.67%

366

366

6.67%

Solution: Taking base 7 antilog on both sides,

log7x = 3

=>X = 73

=>X = 343 7. Solve: log √9 /log 9 zero

zero

6.25%

1/2

1/2

75%

1

1

6.25%

1/3

1/3

12.5%

Solution: (log 91/2 )/  log 9

= 1/2(log 9/ log 9)

= 1/2 8. Solve : logx√3 = 1/2 3

3

81.25%

2

2

6.25%

4

4

6.25%

6

6

6.25%

Solution: On evaluating the given equation,

x >0 , x 1

x1/2 = √3

x1/2 = √3
squaring both side

(x1/2)2 = √32

x = 3 9. Solve: log6x3 = 18 46656

46656

28.57%

4/9

4/9

28.57%

223456

223456

35.71%

4/6

4/6

7.14%

Solution: x3 = 618

x = 66

x = 46656 10. Prove : log636  = 3x 2/3

2/3

66.67%

4/5

4/5

6.67%

6/7

6/7

20%

6/8

6/8

6.67%

Solution: log636 = log6(6)2

2log66 = 2 ( logaa = 1)

2 = 3x

x = 2/3  ×