• $\log _a a=1$

• $\left(\because \log _a a=x \Rightarrow a^x=a^1 \Rightarrow x=1\right)$

• $\log _{1 / a} a=-1$

• $\left(\because \log _{1 / a} a=x \Rightarrow\left(\frac{1}{a}\right)^x=a^1 \Rightarrow x=-1\right)$

• $\log _{a} 1 =0$

• $\left(\because \log _\alpha 1=x \Rightarrow a^x=1=a^0 \Rightarrow x=0\right)$

• $a^{\log _a b}=b$

• Assume $a>0, \quad a \neq 1, \quad m>0, n>0, \quad c>0, c \neq 1$

• 1. $\log _a(m n)=\log _a m+\log _a n \quad$ (Multiplication law)
• 2. $\log _a(m / n)=\log _a m-\log _a n \quad$ (Quotient law)
• 3. $\log _a m=\frac{\log _c m}{\log _c a} \quad$ (Change of base)
• 4. $\log _a\left(m^n\right)=n \log _a m$ (Power law):

• 1. Multiplication law

• $\log _a(m n) =\log _a m+\log _a n$

• $\log _a m =x(\text { say })$

• $a^x =m$,$\log _a n=y \text { (say) }$

• ${a y}=n$ ,$m n =a^x a^y$ ,$m n =a^{x+y}$

• $\log _a m n=x+y$,$\log _a(m n)=\log _a m+\log _a n$,$\log _a(12)=\log _a(4 \times 3)=\log _a 4+\log _a 3$ .

Remark

• $\log _a(m n p q)=\log _a m+\log _a n+\log _a p+\log _a q$

• $\log _a({m n}{p q})=\log _a({m n})+\log _a(p q)$

• 2. Quotient law

• $\log _a\left(\frac{m}{n}\right)=\log _a m-\log _a n \text {. }$,$\log _a m=x(\text { say }) \quad$

• $a^x=m$,$\frac{m}{n}=\frac{a^x}{a^y}=a^{x-y}$

• $\log _a n=y(\text { say })$,$a y=n$ ,$\frac{a^x}{a^y}=a^x(\frac{1}{a^y})=a^x a^{-y}$,$=a^{x - y}$

• $\frac{m}{n}=a^{x-y}$

• $\log _a\left(\frac{m}{n}\right) =x-y$

• $=\log _a m-\log _a n$

• $\log _a\left(\frac{m}{n p q}\right) =\log _a m-\log _a(n p q)$

• $=\log _a m-\log _a n-\log _a p-\log _a q$

• $\log _a\left(\frac{3}{15 \times 4}\right)=\log _{a^3} 3-\log _a 15-\log _a 4$

• 3. Change of base.

• $\log _a m=\frac{\log _c m}{\log _c a}$,$\left(\log _{\underline{a}} m\right)\left(\log _c a\right)=\log _c m$

• $\log _a m=x(\text { say })$ ,$\underline{a}^x=m$ ,$\quad \log _c a=y(\text { say })$,$\quad c^y=a, .$

• $\quad \log _c m=z(\text { say })$

• $\quad c^z=m$

• $(c^y)^x = m = c^z$

• $c^{x y} =c^z$

• $\Rightarrow x y =z$,$\quad\left(\log _a r=\log _a s \Leftrightarrow r=s\right)$

• $\left(\log _a m\right)\left(\log _c a\right)=\log _c m$,$\log _x a=\frac{\log _a a}{\log _a x}=\frac{1}{\log _a x}$

• $\log _x a=\frac{1}{\log _a x}$

• $a=1 \quad \log _x 1=\frac{1}{\log _1 x} \rightarrow \text { (Not defined) }$

• 4. Power law

• $\log _a\left(m^n\right)=n \log _a m \text {. }$

• $\log _a m =x(\text { say }) \quad$

• $a^x=m \ldots (*)$

• $\log _{a}\left(m^n\right)=y(\text { say })$

• $a^y=m^n \ldots (**)$

• from equation $(*)$ and $( * *)$

• $a^y=\left(a^x\right)^n =a^{n x}$

• $\therefore y=n x$

• $\log _a\left(m^n\right)=n \log _a m$

• $\log _2 4-\log _2\left(2^2\right)=2 \log _2 2=2 . [\log _a a=1]$

• 1. Simplify

• $\frac{1}{2} \log _5 36+2 \log _5 7-\frac{1}{2} \log _5 12$

• $=\log _5(36)^{1 / 2}+\log _5 7^2-\log _5(12)^{1 / 2} \quad\left(\log _a\left(m^n\right)=n \log _a m\right)$

• $=\log _5\left(\frac{6 \times 49}{2 \sqrt{3}}\right)=\log _5(49 \sqrt{3}) \text {. }$

• (Multiplication & Quotient law)

• 2. Evaluate

• $\log _5\left(\frac{\sqrt[4]{25}}{625}\right) \quad \sqrt[4]{25}=(25)^{1 / 4} \text {Power law}$

• $= \log _5\left((25)^{1 / 4}\right)-\log _5 625 \text { (Quotient law })$

• $= \frac{1}{4} \log _5 25-\log _5 625$

• $= \frac{1}{4} \log _5\left(5^2\right)-\log _5\left(5^4\right)$

• $= \frac{2}{4} \log _5 5-4 \log _5 5$

• $= \left(\frac{2}{4}-4\right) \log _5 5 \quad \quad \left(\log _a a=1\right)$

• $= \left(\frac{2}{4}-4\right)$

• $=\frac{-14}{4}$

• $=\frac{-7}{2}$

• 3. Evaluate.

• $\log _{10} 2+16 \log _{10}\left(\frac{16}{15}\right)+12 \log _{10}\left(\frac{25}{24}\right)+7 \log _{10}\left(\frac{81}{80}\right)$

• $=\log _{10} 2+\log _{10}\left(\frac{16}{15}\right)^{16}+\log _{10}\left(\frac{25}{24}\right)^{12}+\log _{10}\left(\frac{81}{80}\right)^7 \quad$
• $\text { (Power law) }$
• $\text { (Quotient law) }$
• $\text { (Multiplication) }$

• $=\log _{10}\left(2 \times\left(\frac{16}{15}\right)^{16} \times\left(\frac{25}{24}\right)^{12} \times\left(\frac{81}{80}\right)^7\right)$

• $=\log _{10}\left(\frac{2 \times\left(2^4\right)^{16} \times\left(5^2\right)^{12} \times\left(3^4\right)^7}{3^{16} \times 5^{16} \times 3^{12} \times\left(2^3\right)^{12} \times\left(2^4\right)^7 \times 5^7}\right)$

• $=\log _{10}\left(\frac{2 \times 2^{64} \times 5^{24} \times 3^{28}}{3^{28} \times 5^{23} \times 2^{64}}\right)$

• $=\log _{10}(2 \times 5)=\log _{10} 10=1$ .