If $a$ is a positive real number, other than 1 and $x$ is a rational number such that $a^{x}=N$, then we say that logarithm of $N$ to base $a$ is $x$ or $x$ is the logarithm of $N$ to base $a$,

written as $\log _{a} N=x$.

Thus, $a^{x}=N \Leftrightarrow \log _{a} N=x$.

- Domain : $(0, \infty)$

- Range : $(-\infty, \infty)$

- Period : Non-periodic

- Nature : Neither odd nor even

$\log _{a} N$ is defined only when

(a) $N>0$

(b) $a>0$

(c) $a \neq 1$

(d) For a given value of $N, \log _{a} N$ will give us a unique value.

(e) Logarithm of zero does not exist.

(f) Logarithm of negative real numbers are not defined in the system of real numbers.

This is known as the fundamental logarithmic identity.

Let $M$ and $N$ are arbitrary positive numbers, $a>0, a \neq 1, b>0$, $b \neq 1$

(a) $\log _{a}(M . M)=\log _{a} M+\log _{a} N$

(b) $\log _{a}(M / M)=\log _{a} M-\log _{a} N$

(c) $\log _{a} M^{b}=b \cdot \log _{a} M$

(d) $\log _{b} M=\frac{\log _{a} M}{\log _{a} b}$ (Base changing theorem)

(e) $\log _{a^{b}} m=\frac{1}{b} \log _{a} m$

(f) $a^{\log _{b} c}=c^{\log _{b} a}$

(g) $\log _{x^{a}} y^{b}=\frac{b}{a} \log _{x} y$

(h) $\log _b a \cdot \log _a b=1 \Leftrightarrow \log _b a=1 / \log _a b$.

(i) $\log _b a \cdot \log _c b \cdot \log _a c=1$

(j) $\log _y x \cdot \log _z y \cdot \log _a z=\log _a x$.

(k) $e^{\log a^x}=a^x$

(I) $\log _e a=2.303 \log _{10} a$

(m) $\log _{10} a=0.434 \log _e a$

The positive real number ' $n$ ' is called the antilogarithm of a number ' $m$ ' if $\log n=m$

Thus, $\log n=m\iff n=$ antilog $m$

Example:

$\log 100=2\iff$ antilog $2=100$

Let '$a$' is a real number such that

(a) If $a>1$, then ${\log }_{a}x>{\log }_{a}y$ $\Rightarrow x>y$

(b) If $a>1$, then ${\log }_{\alpha }x<\alpha$ $\Rightarrow 0<x<a^{\alpha }$

(c) If $a>1$, then ${\log }_{a}x>\alpha$ $\Rightarrow x>a^{\alpha }$

(d) If $0<a<1$, then ${\log }_{\alpha }x>{\log }_{a}y$ $\Rightarrow 0<x<y$

(e) If $0<a<1$, then ${\log }_{\alpha }x<\alpha$ $\Rightarrow x>a^{\alpha }$