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].

● 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

From Mathematics Logarithms to HOME PAGE