Logarithm and Modulus Function Formulas

You will never find the concept of Logarithm and Modulus Function and their related work difficult with our Logarithm and Modulus Function Formulae List. Take a quick look at the Formulae Sheet & Tables regarding the concept and learn how to solve logarithms and modulus related problems. Start grasping the Formulas and solve your problems at a faster pace.

1. Definition

a^x = N, x is called the Logarithm of N to the base a. It is also designated as log_aN.
So log_aN = x; a^x = N, a > 0, a ≠ 1 & N > 0
Note:

• The Logarithm of a number is unique i.e. No number can have two different log to a given base.
• From the definition of the Logarithm of the number to a given base ‘a’. a^log_aN = N, a ≠ 1, & N > 0 is known as the fundamental logarithmic identity.
• log_ea = log₁₀a . log_e10 or log₁₀a = \(\frac{\log _{e} a}{\log _{e} 10}\) = 0.434log_ea

2. Properties of Logarithms

Let M and N arbitrary positive number such that a > 0, a ≠ 1, b > 0, b ≠ 1 then

• log_a MN = log_a M + log_a N
• log_a\(\frac{\mathrm{M}}{\mathrm{N}}\) = log_a M – log_a N
• log_aN^α = α log_aN (α any real no;)
• log_a^β N^α = \(\frac{\alpha}{\beta}\) log^aN (α ≠ 0, β ≠ 0)
• log^aN = \(\frac{\log _{b} N}{\log _{b} a}\)
• log^b a. log^a b = 1 ⇒ log^b a = \(\frac{1}{\log _{a} b}\)
• e^lnax = a^x

3. Logarithmic Inequality

Let a is real number such that

• For a > 1 the inequality log_ax > log_a y & x > y are equivalent.
• If a > 1 then log_ax < α ⇒ 0 < x < a^α
• If a > 1 then log_a x > a ⇒ x > a^α
• For 0 < a < 1 the inequality 0 < x < y & log_ax > log_a y are equivalent
• If 0 < a < 1 then log_a x < a ⇒ x > a^α

4. Important Discussion

(i) Given a number N, Logarithms can be expressed as logi0N = Integer + fraction (+ve)

• The mantissa part of log of a number is always kept positive.
• If the characteristics of log₁₀ N be n then the number of digits in N is (n + 1)
• If the characteristics of log₁₀N be (-n) then there exists (n – 1) number of zeros after decimal point of N.

(ii) If the no. & the base are on the same side of the unity, then the logarithm is positive; and If the no. and the base are on different side of unity. Then the logarithm is negative.

5. Modulus Function

Definition: Modulus of a number
Modulus of a number is defined as a (denoted by |a|)
|a| = \(\left\{\begin{array}{lll}a & \text { if } & a>0 \\0 & \text { if } & a=0 \\-a & \text { if } &a<0\end{array}\right\}\) Basic properties of modulus

• |ab| = |a| |b|
• \(\left|\frac{a}{b}\right|=\frac{|a|}{|b|}\) where b ≠ 0
• |a + b| ≤ |a| + |b|
• |a – b| ≥ |a| – |b| equality holds if ab ≥ 0
• If a > 0
  • |x| = a ⇒ x = ± a
  • |x| = -a ⇒ No solution
  • |x| > a ⇒ x< -a or x > a
  • |x|< a ⇒ -a < x < a (v) |x| > -a ⇒ x ∈ R
  • |x| < -a ⇒ No solution

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