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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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