In this article, we will discuss Logarithm Formulas, Logarithm rules, Logarithm properties, and last but not the least Logarithms table.

**Logarithm Concepts & Rules**

The logarithm of any number to a given base is the index or the power to which the base must be raised in order to equal the given number.

**Example:**

If a^{x}=N then x=log_{a}N

This is read as “log N to the base a”.

In the equation, ‘N’ is a positive number and ‘a’ is a positive number other than 1.

**Points to note:**

- Logarithms to the base 10 are known as common logarithms while logarithms to the base e are known as natural logarithms.
- If a logarithm is given without mentioning the base, it is considered as a common logarithm. For Example, 64=4
^{3 }can be expressed as log_{4}64 = 3. - Logarithms have an integral part and a decimal part. The integral part of the logarithm is called the characteristic and the decimal part of the logarithm is called mantissa. The characteristic of the logarithm of a number is determined by inspection and the mantissa by logarithmic table

** E.g:** log 5123 = 3.709 where 3 is the characteristic and 709 is the mantissa.

- When the number is greater than 1 the characteristic is one less than the number of digits to the left of the decimal point in the given number. When the number is less than 1, the characteristic is 1 more than the number of zeroes between the decimal point and the first significant of the number and it is negative.

**For example**, the characteristic of log 3456.25 will be 3 and characteristic of log0.0045 will be -3.

**How to find the Characteristic & Mantissa of a given logarithm number**

For finding characteristic and mantissa of a number, you need to follow the following steps.

- First, convert that number into its standard form. For e.g., Convert 3071 in its standard form.
- For converting this into standard form, make sure that you have to decimalize this number in such a way that the number after the decimal point is single-digit number ranging from 1–9.
- Like in the above question, the answer would be 3.071⋅10
^{3}. Here the characteristic is the power of 10 like in this case the characteristic or the power of 10 is 3.

Now for finding mantissa, you need to understand one concept:

- Let there be a number n whose standard form is written in the form of t⋅10p. Therefore n=t⋅10p
- Now taking log on both sides we get (here log taken is of base 10)

Log n = log(t⋅10p)

Log n = log t+log(10)p

At last, we got,

log n = log t + p

Here p is known as characteristic and log t known as mantissa

## Logarithms Formulas & Logarithms Properties

- log
_{a}a=1 (log of any number to the same base is 1) - log
_{a}1=0 - log
_{a}(m⋅n)= log_{a}m + log_{a}n - log
_{a}(m/n)=log_{a}m - log_{a}n - log
_{a}(1/m)= - log_{a }m - log
_{a}m^{p}= p x log_{a}m - log
_{a}m = log_{b}m/log_{a}b - a
^{logb}= b^{loga} - log
_{a}b = 1/log_{b}a - log
_{a}mb = log_{a}b/log_{am}

**Note:** The base of logarithm can never be equal to 1, i.e log_{1}x is undefined.

**Logarithms Examples**

**1) Use in finding the number of digits of a number**

To find the number of digits in very long large numbers, e.g. 3^{19} or 4^{8}^{12}, the Logarithms formulas are useful.

Values of log1, log 2, log 3 ........to log 9 vary from 0 to 1. Similarly, values of log 10 to log100 vary from 1 to 2. It can be observed that values of logarithms of all 2 digit numbers are of form 1.xxxxxxx, values of logarithms of all 3 digit numbers are of the form 2.xxxxxx and so on. Therefore, a number with a logarithm value of n.xxxxx will have n+1 digits.

For example, 48^{12}can be written as 2^{48} x 3^{12}

The logarithm of this number is 48log2 + 12log3 = 48 x 0.301 + 12 x 0.4771 = 20.17

Therefore, the number 48^{12} has 21 digits.

**Question 1:** If log 2 = 0.30103, Find the number of digits in 256 is

**Solution:**

log(256)

Using Logarithms Formulas

log(256) =56*0.30103 =16.85768.**Its characteristic is 16.**

Hence, the number of digits in **2**^{56}** is 17.**

**Question 2: Find the number of digits in 8**^{10}** ? (Given that log****10**** 2 = 0.3010)**

8^{10} = (2^{3})^{10}

Using Logarithms Formulas

∴ Required answer = [30 log_{10} 2 + 1]

= [30 x 0.3010] + 1

= 9.03 + 1

= 9 + 1

= 10