Skip to main content

Logarithm - Notes, Concept and All Important Formula

LOGARITHM

LOGARITHM OF A NUMBER :

The logarithm of the number \(\mathrm{N}\) to the base ' \(\mathrm{a}\) ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number \(\mathrm{N}\). This number is designated as \(\log _{\mathrm{a}} \mathrm{N}\).

(a) \(\log _{a} \mathrm{~N}=\mathrm{x}\), read as \(\log\) of \(\mathrm{N}\) to the base \(\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}\).
If \(a=10\) then we write \(\log N\) or \(\log _{10} \mathrm{~N}\) and if \(\mathrm{a}=e\) we write \(\ln N\) or \(\log _{e} \mathrm{~N}\) (Natural log)
(b) Necessary conditions : \(\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1\)
(c) \(\log _{a} 1=0\)
(d) \(\log _{a} a=1\)
(e) \(\log _{1 / a} a=-1\)
(f) \(\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0\)
(g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{x}, \mathrm{y}> \,\,0\)
(h) \(\log _{a} x^{p}=p \log _{a} x ; \,\, x> \,\,0\)
(i) \(\log _{a^{9}} x=\dfrac{1}{q} \log _{a} x ; \,\, x> \,\,0\)
(j) \(\log _{a} x=\dfrac{1}{\log _{x} a} ; \,\, x> \,\,0, x \neq 1\)
(k) \(\log _{a} x=\log _{b} x / \log _{b} a ; \,\, x> 0,\) \( a, b> 0,\) \( b \neq 1,\) \( a \neq 1\)
(l) \(\log _{a} b \cdot \log _{b} c . \log _{c} \mathrm{~d}=\log _{a} \mathrm{~d} ; \,\, \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}> \,\,0, \neq 1\)
(m) \(a^{\log _{a} x}=x ; \,\, a> \,\,0, a \neq 1\)
(n) \(a^{\log _{b} c}=c^{\log _{b} a} ; \,\, a, b, c> \,\,0 ; \,\, b \neq 1\)
(o) \(\log _{a} x< \,\,\log _{a} y \Leftrightarrow\left[\begin{array}{ll}x< \,\,y & \,\, \text { if } \quad a> \,\,1 \\ x> \,\,y & \,\, \text { if } \quad 0< \,\,a< \,\,1\end{array}\right.\)
(p) \(\log _{a} x=\log _{a} y \Rightarrow x=y ;\)\( \,\, x, y> 0 ; \,\, \)\(a> \,\,0, a \neq 1\)
(q) \(e^{\operatorname{lna}^{x}}=a^{x}\)
(r) \(\log _{10} 2=0.3010 ;\)\( \,\, \log _{10} 3=0.4771 ;\)\( \,\, \ln 2=0.693,\)\( \ln 10=2.303\)
(s) If \(a> \,\,1\) then \(\log _{a} x< \,\,p \Rightarrow 0< \,\,x< \,\,a^{p}\)
(t) If \(a> \,\,1\) then \(\log _{a} x> \,\,p \Rightarrow x> \,\,a^{p}\)
(u) If \(0< \,\,a< \,\,1\) then \(\log _{a} x< \,\,p \Rightarrow x> \,\,a^{p}\)
(v) If \(0< \,\,a< \,\,1\) then \(\log _{a} x> \,\,p \Rightarrow 0< \,\,x< \,\,a^{p}\)
Labels:

Comments

Post a Comment

Popular posts from this blog

Determinant - Notes, Concept and All Important Formula

DETERMINANT 1. MINORS : The minor of a given element of determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of \(a _{1}\) in \(\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right|\) is \(\left|\begin{array}{ll} b _{2} & c _{2} \\ b _{3} & c _{3}\end{array}\right| \&\) the minor of \(b_{2}\) is \(\quad\left|\begin{array}{ll}a_{1} & c_{1} \\ a_{3} & c_{3}\end{array}\right|\) . Hence a determinant of order three will have " 9 minors". 2. COFACTORS : If \(M _{ ij }\) represents the minor of the element belonging to \(i ^{\text {th }}\) row and \(j ^{\text {th }}\) column then the cofactor of that element : \(C_{i j}=(-1)^{i+j} \cdot M_{i j}\) Important Note : Consider \(\Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} ...
Post a Comment
Read more

Circle- Notes, Concept and All Important Formula

CIRCLE 1. DEFINITION : A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle. All Chapter Notes, Concept and Important Formula 2. STANDARD EQUATIONS OF THE CIRCLE : (a) Central Form: If \((\mathrm{h}, \mathrm{k})\) is the centre and \(\mathrm{r}\) is the radius of the circle then its equation is \((\mathbf{x}-\mathbf{h})^{2}+(\mathbf{y}-\mathbf{k})^{2}=\mathbf{r}^{2}\) (b) General equation of circle : \(\mathbf{x}^{2}+\mathbf{y}^{2}+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{c}=\mathbf{0}\) , where \(g, \mathrm{f}, c\) are constants and centre is \((-g,-f)\) i.e. \(\left(-\frac{\text { coefficient of } \mathrm{x}}{2},-\frac{\text { coefficient of } \mathrm{y}}{2}\right)\) and radius \(r=\sqrt{g^{2}+f^{2}-c}\) Note : The general quadratic equation in \(\mathrm{x}\) and \(\mathrm{y}\) , \(a x^{2}+b y^{2}+2 ...
Post a Comment
Read more

Definite Integration - Notes, Concept and All Important Formula

DEFINITE INTEGRATION Table Of Contents 1. (a) The Fundamental Theorem of Calculus, Part 1: If \(\mathrm{f}\) is continuous on \([\mathrm{a}, \mathrm{b}]\) , then the function \(\mathrm{g}\) defined by \(g(x)=\displaystyle \int_{a}^{x} f(t) d t,\)   \(  a \leq x \leq b\) is continuous on \([\mathrm{a}, \mathrm{b}]\) and differentiable on \((\mathrm{a}, \mathrm{b})\) , and \(g^{\prime}(\mathrm{x})=\mathrm{f}(\mathrm{x})\) . (b) The Fundamental Theorem of Calculus, Part 2: If f is continuous on \([a, b]\) , then \(\displaystyle \int_{a}^{b} f(x) d x=F(b)-F(a)\) where \(F\) is any antiderivative of \(\mathrm{f}\) , that is, a function such that \(\mathrm{F}^{\prime}=\mathrm{f}.\) Note : If \(\displaystyle \int_{a}^{b} f(x) d x=0 \Rightarrow\) then the equation \(f(x)=0\) has atleast one root lying in \((a, b)\) provided \(f\) is a continuous function in \((a, b)\) . All Chapter Notes, Concept and Important Formula 2. Representation of Definite Int...
Post a Comment
Read more