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**Logarithm **

The logarithm of a number to a given base is the index or the power to which the base must be raised to obtain that number.

If a^{m} = b (where b > 0, a ≠ 1, a > 0), then the exponent m is said to be logarithm of b to the base a.

So, if a^{m} = b (Exponent form)

Then, m = log_{a} b (Logarithm form)

**Example 1. Express in logarithmic form: (i) 2 ^{3} (ii) x^{m} = y**

**Solution : **2^{3} = 8 ⇒ log_{2}8 = 3

x^{m} = y ⇒ log_{x}y = m

**Note: **

- Logarithm of any quantity to the same base is 1.
- Logarithm of 1 to any base is zero

**Types of Logarithm**

1. Logarithms to the base “e” (exponential constant) are called natural logarithms.

2. Logarithms to the base “10” are called common logarithms

**Note: I**f the base is not given, it can be assumed to be 10

**Laws of Logarithm**

**1. log _{a}(mn) = log_{a} m + log_{a} n**

Logarithm of a product of two numbers can be written as sum of logarithms of the individual numbers.

**Example 1.** **If log _{10} 2 = 0.3010, log_{10} 3 = 0.4771, find log_{10} 6.**

**Solution : **log_{10} 6 = ** **log_{10} (2x3)

log_{10} 2= ** **log_{10} 3

= 0.3010 + 0.4771 = 0.7781

**Example 2. Simplify log 2 + log 3 + log 4**

**Solution : l**og 2 + log 3 + log 4 = log (2 × 3 × 4) = log 24

**2. **

**Logarithm of a division of two numbers can be written as difference of logarithms of the individual numbers.**

**Example: **

**3. (i) log _{a}m^{n} = n log_{a}m**

**(ii) **

Logarithm of the n^{th} power of a number is n times the logarithm of that number.

Logarithm of a number m to the base a raised to power n is equal to 1/n times the logarithm of m to the base a.

**Example 1 : **** If log _{10} 2 = 0.3010, find log_{10} 8.**

**Solution : **log_{10}8 = log_{10}2 = 3 log_{10} 2 = 3 × 0.3010 = 0.9030

**Example 2.** ** Find the value of 1/2log _{10} = 100**

**Solution : **

**4. **

**Example : **

**Solution : **

= log_{10}(2)

**5. **

We can write any logarithm as a division of two logarithms (of the number and the base) taken at any common base. This theorem is very useful in solving problems having logarithms with many different bases.If we put m = a in the above result, we get another important result which is log_{n} a × log_{a} n = 1

**Example : Find the value of log _{3} 7 x log_{49} 243**

**Solution : ** To simplify, we will make use of change of base theorem and convert each term into a logarithm with common base of 10.

**6. If log _{a} m = x, then**

a. log_{(1/a)} m = -x

b.

c.

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