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 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 aⁿ = 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,

aⁿ = b ⇔ log_a b = n.

aⁿ is called the exponential form and log_a 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 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, a⁰ = 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 a¹ = 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].

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