The logarithm of the number $\mathrm{N}$$\mathrm{N}$N\mathrm{N}$\mathrm{N}$ to the base ‘ $\mathrm{a}$$\mathrm{a}$a\mathrm{a}$\mathrm{a}$ ‘ is the exponent indicating the power to which the base ‘a’ must be raised to obtain the number $N$$N$NN$N$. This number is designated as ${\log }_{a}N$${\log }_{a}N$log_(a)N\log _{a} N${\log }_{a}N$.
(a) ${\log }_a\mathrm{N}=\mathrm{x}$${\log }_a\mathrm{N}=\mathrm{x}$log_(a)N=x\log _{a} \mathrm{~N}=\mathrm{x}${\log }_a\mathrm{N}=\mathrm{x}$, read as $\log$$\log$log\log$\log$ of $\mathrm{N}$$\mathrm{N}$N\mathrm{N}$\mathrm{N}$ to the base $\mathrm{a}\iff {\mathrm{a}}^{\mathrm{x}}=\mathrm{N}$$\mathrm{a}\iff {\mathrm{a}}^{\mathrm{x}}=\mathrm{N}$a<=>a^(x)=N\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}$\mathrm{a}\iff {\mathrm{a}}^{\mathrm{x}}=\mathrm{N}$ If $\mathrm{a}=10$$\mathrm{a}=10$a=10\mathrm{a}=10$\mathrm{a}=10$ then we write $\log \mathrm{N}$$\log \mathrm{N}$log N\log \mathrm{N}$\log \mathrm{N}$ or ${\log }_{10}\mathrm{N}$${\log }_{10}\mathrm{N}$log_(10)N\log _{10} \mathrm{~N}${\log }_{10}\mathrm{N}$ and if $\mathrm{a}=e$$\mathrm{a}=e$a=e\mathrm{a}=e$\mathrm{a}=e$ we write $\ln \mathrm{N}$$\ln \mathrm{N}$ln N\ln \mathrm{N}$\ln \mathrm{N}$ or ${\log }_e\mathrm{N}$${\log }_e\mathrm{N}$log_(e)N\log _{e} \mathrm{~N}${\log }_e\mathrm{N}$ (Natural $\log$$\log$log\log$\log$ )
(b) Necessary conditions: $\mathrm{N}>0;\mathrm{a}>0;\mathrm{a}\ne 1$$\mathrm{N}>0;\mathrm{a}>0;\mathrm{a}\ne 1$N > 0;a > 0;a!=1\mathrm{N}>0 ; \mathrm{a}>0 ; \mathrm{a} \neq 1$\mathrm{N}>0;\mathrm{a}>0;\mathrm{a}\ne 1$
(c) ${\log }_{a}1=0$${\log }_{a}1=0$log_(a)1=0\log _{a} 1=0${\log }_{a}1=0$
(d) ${\log }_{\mathrm{a}}\mathrm{a}=1$${\log }_{\mathrm{a}}\mathrm{a}=1$log_(a)a=1\log _{\mathrm{a}} \mathrm{a}=1${\log }_{\mathrm{a}}\mathrm{a}=1$
(e) ${\log }_{\frac{1}{a}}a=-1$${\log }_{\frac{1}{a}}a=-1$log_((1)/(a))a=-1\log _{\frac{1}{a}} a=-1${\log }_{\frac{1}{a}}a=-1$
(f) ${\log }_a(x.y)={\log }_{a}x+{\log }_{a}y;x,y>0$${\log }_a(x.y)={\log }_{a}x+{\log }_{a}y;x,y>0$log_(a)(x.y)=log_(a)x+log_(a)y;x,y > 0\log _{a}(x . y)=\log _{a} x+\log _{a} y ; x, y>0${\log }_a(x.y)={\log }_{a}x+{\log }_{a}y;x,y>0$
(g) ${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y;x,y>0$${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y;x,y>0$log_(a)((x)/(y))=log_(a)x-log_(a)y;x,y > 0\log _{a}\left(\frac{x}{y}\right)=\log _{a} x-\log _{a} y ; x, y>0${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y;x,y>0$
(h) ${\log }_a{x}^p=p{\log }_{a}x;x>0$${\log }_a{x}^p=p{\log }_{a}x;x>0$log_(a)x^(p)=plog_(a)x;x > 0\log _{a} x^{p}=p \log _{a} x ; x>0${\log }_a{x}^p=p{\log }_{a}x;x>0$
(i) ${\log }_{a^q}x=\frac{1}{q}{\log }_{a}x;x>0$${\log }_{a^q}x=\frac{1}{q}{\log }_{a}x;x>0$log_(a^(q))x=(1)/(q)log_(a)x;x > 0\log _{a^{q}} x=\frac{1}{q} \log _{a} x ; x>0${\log }_{a^q}x=\frac{1}{q}{\log }_{a}x;x>0$
(j) ${\log }_{a}x=\frac{1}{{\log }_{x}a};x>0,x\ne 1$${\log }_{a}x=\frac{1}{{\log }_{x}a};x>0,x\ne 1$log_(a)x=(1)/(log_(x)a);x > 0,x!=1\log _{a} x=\frac{1}{\log _{x} a} ; x>0, x \neq 1${\log }_{a}x=\frac{1}{{\log }_{x}a};x>0,x\ne 1$
(k) ${\log }_{a}x={\log }_{b}x/{\log }_{b}a;x>0,a,b>0,b\ne 1,a\ne 1$${\log }_{a}x={\log }_{b}x/{\log }_{b}a;x>0,a,b>0,b\ne 1,a\ne 1$log_(a)x=log_(b)x//log_(b)a;x > 0,a,b > 0,b!=1,a!=1\log _{a} x=\log _{b} x / \log _{b} a ; x>0, a, b>0, b \neq 1, a \neq 1${\log }_{a}x={\log }_{b}x/{\log }_{b}a;x>0,a,b>0,b\ne 1,a\ne 1$
(l) ${\log }_{a}b\cdot {\log }_{b}c\cdot {\log }_{c}d={\log }_{a}d;a,b,c,d>0,\ne 1$${\log }_{a}b\cdot {\log }_{b}c\cdot {\log }_{c}d={\log }_{a}d;a,b,c,d>0,\ne 1$log_(a)b*log_(b)c*log_(c)d=log_(a)d;a,b,c,d > 0,!=1\log _{a} b \cdot \log _{b} c \cdot \log _{c} d=\log _{a} d ; a, b, c, d>0, \neq 1${\log }_{a}b\cdot {\log }_{b}c\cdot {\log }_{c}d={\log }_{a}d;a,b,c,d>0,\ne 1$
(m) $a^{{\log }_{a}x}=x;a>0,a\ne 1$$a^{{\log }_{a}x}=x;a>0,a\ne 1$a^(log_(a)x)=x;a > 0,a!=1a^{\log _{a} x}=x ; a>0, a \neq 1$a^{{\log }_{a}x}=x;a>0,a\ne 1$
(n) $a^{{\log }_{b}c}=c^{{\log }_{b}a};a,b,c>0;b\ne 1$$a^{{\log }_{b}c}=c^{{\log }_{b}a};a,b,c>0;b\ne 1$a^(log_(b)c)=c^(log_(b)a);a,b,c > 0;b!=1a^{\log _{b} c}=c^{\log _{b} a} ; a, b, c>0 ; b \neq 1$a^{{\log }_{b}c}=c^{{\log }_{b}a};a,b,c>0;b\ne 1$
(o) ${\log }_{a}x<{\log }_{a}y\iff [\begin{array}{llc}x<y& \text{if}& a>1\\ x>y& \text{if}& 0<a<1\end{array}$${\log }_{a}x<{\log }_{a}y\iff \left[\begin{array}{llc}x<y& \text{ if }& a>1\\ x>y& \text{ if }& 0<a<1\end{array}\right.$log_(a)x < log_(a)y<=>[[x < y,” if “,a > 1],[x > y,” if “,0 < a < 1]:}\log _{a} x<\log _{a} y \Leftrightarrow\left[\begin{array}{llc}x<y & \text { if } & a>1 \\ x>y & \text { if } & 0<a<1\end{array}\right.${\log }_{a}x<{\log }_{a}y\iff [\begin{array}{llc}x<y& \text{ if }& a>1\\ x>y& \text{ if }& 0<a<1\end{array}$
(p) ${\log }_a\mathrm{x}={\log }_{\mathrm{a}}\mathrm{y}\Rightarrow \mathrm{x}=\mathrm{y};\mathrm{x},\mathrm{y}>0;\mathrm{a}>0,\mathrm{a}\ne 1$${\log }_a\mathrm{x}={\log }_{\mathrm{a}}\mathrm{y}\Rightarrow \mathrm{x}=\mathrm{y};\mathrm{x},\mathrm{y}>0;\mathrm{a}>0,\mathrm{a}\ne 1$log_(a)x=log_(a)y=>x=y;x,y > 0;a > 0,a!=1\log _{a} \mathrm{x}=\log _{\mathrm{a}} \mathrm{y} \Rightarrow \mathrm{x}=\mathrm{y} ; \mathrm{x}, \mathrm{y}>0 ; \mathrm{a}>0, \mathrm{a} \neq 1${\log }_a\mathrm{x}={\log }_{\mathrm{a}}\mathrm{y}\Rightarrow \mathrm{x}=\mathrm{y};\mathrm{x},\mathrm{y}>0;\mathrm{a}>0,\mathrm{a}\ne 1$
(q) $e^{\ln a^x}=a^x$$e^{\ln a^x}=a^x$e^(ln a^(x))=a^(x)e^{\ln a^{x}}=a^{x}$e^{\ln a^x}=a^x$
(r) ${\log }_{10}2=0.3010;{\log }_{10}3=0.4771;\ln 2=0.693,\ln 10=2.303$${\log }_{10}2=0.3010;{\log }_{10}3=0.4771;\ln 2=0.693,\ln 10=2.303$log_(10)2=0.3010;log_(10)3=0.4771;ln 2=0.693,ln 10=2.303\log _{10} 2=0.3010 ; \log _{10} 3=0.4771 ; \ln 2=0.693, \ln 10=2.303${\log }_{10}2=0.3010;{\log }_{10}3=0.4771;\ln 2=0.693,\ln 10=2.303$
(s) If $a>1$$a>1$a > 1a>1$a>1$ then ${\log }_{a}x<p\Rightarrow 0<x<a^p$${\log }_{a}x<p\Rightarrow 0<x<a^p$log_(a)x < p=>0 < x < a^(p)\log _{a} x<p \Rightarrow 0<x<a^{p}${\log }_{a}x<p\Rightarrow 0<x<a^p$
(t) If $a>1$$a>1$a > 1a>1$a>1$ then ${\log }_{a}x>p\Rightarrow x>a^p$${\log }_{a}x>p\Rightarrow x>a^p$log_(a)x > p=>x > a^(p)\log _{a} x>p \Rightarrow x>a^{p}${\log }_{a}x>p\Rightarrow x>a^p$
(u) If $0<a<1$$0<a<1$0 < a < 10<a<1$0<a<1$ then ${\log }_{a}x<p\Rightarrow x>a^p$${\log }_{a}x<p\Rightarrow x>a^p$log_(a)x < p=>x > a^(p)\log _{a} x<p \Rightarrow x>a^{p}${\log }_{a}x<p\Rightarrow x>a^p$
(v) If $0<a<1$$0<a<1$0 < a < 10<a<1$0<a<1$ then ${\log }_{a}x>p\Rightarrow 0<x<a^p$${\log }_{a}x>p\Rightarrow 0<x<a^p$log_(a)x > p=>0 < x < a^(p)\log _{a} x>p \Rightarrow 0<x<a^{p}${\log }_{a}x>p\Rightarrow 0<x<a^p$