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 = πr2h = 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 ax = 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 = loga M.

Hence, if ax = M then x = loga M;

conversely, if x = loga M then ax = M.


If ‘a’ is a positive real number (except 1), n is any real number and an = b, then n is called the logarithm of b to the base a. It is written as loga b (read as log of b to the base a). Thus,

an = b ⇔ loga b = n.


an is called the exponential form and loga b = n is called the logarithmic form.

For example:

32 = 9 ⇔ log3 9 = 2

54 = 625 ⇔ log5 625 = 4

70 = 1 ⇔ log7 1 = 0

2-3 = 1/8 ⇔ log2 (1/8) = -3

10-2 = 0.01 ⇔ log10 0.01 = -2

26 = 64 ⇔ log2 64 = 6

3- 4 = 1/34 = 1/81 ⇔ log3 1/81 = -4

10-2 = 1/100 = 0.01 ⇔ log10 0.01 = -2

Notes on basic Logarithm Facts:

1. Since a > 0 (a ≠ 1), an > 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 log10 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 loge x.


6. If ax = - 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, a0 = 1 (a ≠ 0). Therefore, from the definition, we have, loga 1 = 0.


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

Proof: Since a1 = a. Therefore, loga a = 1.

Note:

From 7 and 8 we say that, loga 1 = 0 and loga a = 1 for any positive real ‘a’ except 1.


9. If x = loga M then a log a M = a

Proof: x = loga M. Therefore, ax = M or, a logaM = M [Since, x = loga M].

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