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,

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

*logarithm tables***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

**. Thus ln x means log**

*ln*_{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**for any positive real ‘a’ except 1.

_{a}a = 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**

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**Common Logarithm and Natural Logarithm**

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