Logarithm Formulas

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Here we are providing logarithm formulas for maths. These formulas are important for students studying in science and commerce stream. These formulas are very important for Physics and Chemistry also. Students are suggested to go through all the formulas. It will help students to solve physics and chemistry problems involving the concept of logarithm.

(a) $$\log _a N=x$$ ⇒ a x = N
It read as log of N to the base ‘a’
If $$\mathrm{a}=10$$ then we write $$\log \mathrm{N}$$ or $$\log _{10} \mathrm{~N}$$ and if $$\mathrm{a}=e$$ we write ln $$\mathrm{N}$$ or $$\log _e \mathrm{~N} \text {(Natural log)} $$ (b) Necessary conditions: $$\mathrm{N}>0 ; \mathrm{a}>0 ; \mathrm{a} \neq 1$$ (c) $$\log _a 1=0$$ (d) $$\log _a a=1$$ (e) $$\log _{\frac{1}{a}} a=-1$$ (f) $$\log _a(x . y)=\log _a x+\log _a y ; x, y>0$$ (g) $$\log _a\left(\frac{x}{y}\right)=\log _a x-\log _a y ; x, y>0$$ (h) $$\log _a x^p=p\log _a x\ ; x>0$$ (i) $$\quad \log _{a^9} x=\frac{1}{q} \log _a x ; x>0$$ (j) $$\log _a x=\frac{1}{\log _x a} ; x>0, x \neq 1$$ (k) $$\log _a x=\log _b x / \log _b a ; \\x>0, a, b>0, b \neq 1, a \neq 1$$ (l) $$\log _a b \cdot \log _b c.\log _c \mathrm{~d}=\log _{\mathrm{a}} \mathrm{d} ;\\ \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}>0, \neq 1$$ (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) $$\log _a x=\log _a y \Rightarrow x=y ; x, y>0 ; a>0, a \neq 1$$ (p) $$e^{\ln ^x}=a^x$$ (r) $$\log _{10} 2=0.3010 ; \\ \log _{10} 3=0.4771 ; \\ \ln 2=0.693, \\ \ln 10=2.303$$

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