Quick Review: Logarithm

Formulas

Any numeral is known as a number. Numbers are of various types. Let us discuss the types of numbers.

• Definition: a^x = b can be represented in logarithmic form as log_a b = x
• log a = x means that 10^x = a .
• 10^{log a} = a (The basic logarithmic identity).
• log (ab) = log a + log b, a > 0, b > 0
• log(a/b) = log a -log b, a > 0, b > 0.
• log aⁿ = n (log a) (Logarithm of a power).
• log_x y = log_my / log_mx (Change of base rule).
• log_x y = 1 / log_y x .
• log_x1 = 0 (x ≠ 0, 1).
• The natural numbers 1, 2, 3,.... are respectively the logarithms of 10, 100, 1000, .... to the base 10.
• The logarithm of "0" and negative numbers are not defined.
• log_b1= 0 (∵ b⁰ = 1)
• log_b b = 1 (∵ b₁ = b)
• y = ln x → e^y = x
• x = e^y = → ln x = y
• x = ln e^x = e^{ln x}
• e^log_b⁡x = x
• log_b b^y = y

Laws of Logarithm

• log_b⁡MN = log_b⁡M + log_b⁡N (where b, M and N are positive real numbers and b ≠ 1)
• log_b⁡(M/N) = log_b⁡M - log_b⁡N (where b, M and N are positive real numbers and b ≠ 1)
• log_b⁡(M^c ) = c log_b⁡M (where b, M and N are positive real numbers and b ≠ 1 and c is any real number)
• log_b⁡M = log⁡M/log⁡b = ln⁡M/ln⁡b = log_k⁡M/log_k⁡b (where b, M and k are positive real numbers and b ≠ 1, k≠1)
• log_b⁡a = 1/log_a⁡b (where b, and a are positive real numbers and b ≠ 1, a≠1)
• If log_b⁡M = log_b⁡N, then M = N (where b, M and N are positive real numbers and b ≠ 1)

Euler's Number

A mathematical consonant e is the base of the natural logarithm, known as Euler's number. It is also known as Napier's consonant.