# Mathematics Logarithms

In mathematics logarithms were developed for making complicated calculations simple.

For example, if a right circular cylinder has radius r = 0.375 meters and height h = 0.2321 meters, then its volume is given by: V = A = πr

^{2}h = 3.146 × (0.375)

^{2} × 0.2321. Use for logarithm tables makes such calculations quite easy. However, even calculators have functions like multiplication; power etc. still, logarithmic and exponential equations and functions are very common in mathematics.

Definition:

If a

^{x} = M (M > 0, a > 0, a ≠ 1), then x (i.e., index of the power) is called the logarithm of the number

**M** to the base

**a** and is written as

x = log_{a} M.

Hence, if a

^{x} = M then x = log

_{a} M;

conversely, if x = log

_{a} M then a

^{x} = M.

If ‘

**a**’ is a positive real number (except 1),

**n** is any real number and

**a**^{n} = b, then

**n** is called the

*logarithm of b to the base a*.
It is written as log

_{a} b (read as log of b to the base a).
Thus,

**a**^{n} = b ⇔ log_{a} b = n.
a

^{n} is called the exponential form and log

_{a} b = n is called the logarithmic form.

### For example:

● 3

^{2} = 9 ⇔ log

_{3} 9 = 2

● 5

^{4} = 625 ⇔ log

_{5} 625 = 4

● 7

^{0} = 1 ⇔ log

_{7} 1 = 0

● 2

^{-3} =

^{1}/

_{8} ⇔ log

_{2} (

^{1}/

_{8}) = -3

● 10

^{-2} = 0.01 ⇔ log

_{10} 0.01 = -2

● 2

^{6} = 64 ⇔ log

_{2} 64 = 6

● 3

^{- 4} = 1/3

^{4} = 1/81 ⇔ log

_{3} 1/81 = -4

● 10

^{-2} = 1/100 = 0.01 ⇔ log

_{10} 0.01 = -2

### Notes on basic Logarithm Facts:

**1.** Since a > 0 (a ≠ 1), a

^{n} > 0 for any rational n. Hence logarithm is defined only positive real numbers.

From the definition it is clear that the logarithm of a number has no meaning if the base is not mentioned.

**2.** The above examples shows that the logarithm of a (positive) real number may be negative, zero or positive.

**3.** Logarithmic values of a given number are different for different bases.

**4.** Logarithms to the base a 10 are called

*common logarithms*. Also,

*logarithm tables* assume base 10. If no base is given, the base is assumed to be 10.

**For example:** log 21 means log

_{10} 21.

**5.** Logarithm to the base ‘

**e**’ (where e = 2.7183 approx.) is called

*natural logarithm*, and is usually written as

*ln*. Thus ln x means log

_{e} x.

**6.** If a

^{x} = - M (a > 0, M > 0), then the value of x will be imaginary i.e., logarithmic value of a negative number is imaginary.

**7.** Logarithm of 1 to any finite non-zero base is zero.

**Proof: **We know, a

^{0} = 1 (a ≠ 0). Therefore, from the definition, we have, log

_{a} 1 = 0.

**8.** Logarithm of a positive number to the same base is always 1.

**Proof: **Since a

^{1} = a. Therefore, log

_{a} a = 1.

### Note:

From 7 and 8 we say that,

**log**_{a} 1 = 0 and

**log**_{a} a = 1 for any positive real ‘a’ except 1.

**9.** If x = log

_{a} M then a

^{log a M} = a

**Proof: **
x = log

_{a} M. Therefore, a

^{x} = M or, a

^{log}_{a}^{M} = M [Since, x = log

_{a} M].

**●** **Mathematics Logarithm**

**Mathematics Logarithms**

**Convert Exponentials and Logarithms**

**Logarithm Rules or Log Rules**

**Solved Problems on Logarithm**

**Common Logarithm and Natural Logarithm**

**Antilogarithm**

**Logarithms **

**11 and 12 Grade Math**

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