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 $\log _{a} x> \,\,p \Rightarrow 0< \,\,x< \,\,a^{p}$

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All Chapter Notes, Concept and Important Formula

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