# Logarithm and Modulus Function Formulas

Logarithm and Modulus Function related problems can be tricky. Don’t get discouraged however. Once you figure out the Logarithm and Modulus Function Formulas they aren’t that bad too. To help you in this we have jotted the Logarithm and Modulus Formulae all at one place that can be of some help during your calculation part. Simplification becomes little simple by applying the Log and Modulus Functions Formulas List provided.

## List of Logarithm and Modulus Function Formulae

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_{a}N.

So log_{a}N = 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
^{logaN}= N, a ≠ 1, & N > 0 is known as the fundamental logarithmic identity. - log
_{e}a = log_{10}a . log_{e}10 or log_{10}a = \(\frac{\log _{e} a}{\log _{e} 10}\) = 0.434log_{e}a

**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
_{a}N^{α}= α log_{a}N (α any real no;) - log
_{aβ}N^{α}= \(\frac{\alpha}{\beta}\) log^{a}N (α ≠ 0, β ≠ 0) - log
^{a}N = \(\frac{\log _{b} N}{\log _{b} a}\) - 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
_{a}x > log_{a}y & x > y are equivalent. - If a > 1 then log
_{a}x < 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
_{a}x > 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 log
_{10}N be n then the number of digits in N is (n + 1) - If the characteristics of log
_{10}N 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