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

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
Post a Comment
Read more

Inverse Trigonometric function - Notes, Concept and All Important Formula

INVERSE TRIGONOMETRIC FUNCTION 1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC FUNCTIONS : (a) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x})\) (b) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x}\) (c) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2)\) , \(\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x}\) (d) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi)\) \(\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x}\) (e) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[0, \pi / 2) \cup(\pi / 2, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x}\) (f) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[-\pi / 2,0) \cup(0, \pi / 2]\) \(\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x}\) All Chapter Notes, Concept and Important Formula 2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS: Property-1 : (i) \(y=\sin ...
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