# Aptitude Logarithm Concepts and Formulas

General rule: The logarithm of any positive real number y, other than 1 when ax = y, then component x is called the logarithm of y to the base a, then x= loga y.

### Important terms:

a) The logarithm of one (1) to any base is zero.

```Loga 1 = 0
```

b) The logarithm of zero (0) to any base greater than unity is -∞.

```Loga  0 = -∞
```

c) The logarithm of any number (a) to the same base is always unity.

```Loga  a = 1
```

d) Let logb x = p, then x= bp

```Or, x = b logb x
```

e) The logarithm of product:

```Loga  (m*n) = loga  m * loga  n
```

f) The logarithm of the fraction:

```loga (m/n)= loga  m - loga  n
```

g) Power formula:

```Loga  mn = n loga  m.
```

h) Logb a = logc a / logc b

i) Logb c = 1 / logc b

j) Logb a = logc a * logb c

k) Logxn (ym) = mloga y/ nloga x

Common logarithm: Logarithms to the base 10 are known as a common logarithms.

Therefore, Log10 10 = 1 is known as a common logarithm.

### Characteristic and mantissa:

Every logarithm has two parts: the whole of the integer part and the fraction or decimal parts. The integer part is called the characteristics, and decimal part is called the mantissa.

### There are two rules for characteristics:

#### i. To find the characteristics of a number greater than one.

Characteristic is 1 less than the number of digits to the left of the decimal point in the given numbers.

#### ii. To find the characteristic of a number less than one.

Characteristic is 1 more than the number of zeros between the decimal point and the 1st significant digit of the number. The number is represented by a bar because of negative.

For example:

Number Characteristic Number characteristic
6.125 0 (only one digit (6) before the decimal point that?s why characteristic =1-1 =0) 0.6125 ͞1 (there is no zero between decimal point, and the first significant digit (here first significant bit is 6). So the characteristic will be 0+1 = 1)
65.23 2-1 = 1 0.06125 1+1 = ͞2
652.3 3-1 = 2 0.006125 2+1 = ͞3
6523 4-1 = 3 0.0006125 3+1 = ͞4

Aptitude Logarithm Test Paper 1
Aptitude Logarithm Test Paper 2
Aptitude Logarithm Test Paper 3
Aptitude Logarithm Test Paper 4

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