Indices and Logarithms

Laws of Indices

 
Law 1 \(a^p\times a^q=a^{p+q}\) \(3^2\times3^4=3^6\)
Law 2 \(\frac{a^p}{a^q}=a^{p-q}\) \(\frac{7^6}{7^3}=7^3\)
Law 3 \((a^p)^q=a^{pq}\) \((4^2)^3=4^6\)
Law 4 \(a^0=1\) \(6^0=1\)
Law 5 \(a^{\frac{1}{q}}=(\sqrt[q]{a})^p\) \(27^{\frac{1}{3}}=\sqrt[3]{27}=3\) \(\quad\quad\quad[\sqrt{a}=a^\frac{1}{2}]\)
Law 6 \(a^\frac{p}{q}=(\sqrt[q]{a})^p\) \(4^\frac{3}{2}=(\sqrt[2]{4})^3=8\)
Law 7 \(a^{-p}=\frac{1}{a^p}\) \(3^{-2}=\frac{1}{3^2}=\frac{1}{9}\)
Law 8 \((ab)^p=(a^p)(b^p)\) \((2x)^6=(2)^6(x)^6\)
Law 9 \((\frac{a}{b})^p=\frac{a^p}{b^p}\) \(((\frac{4}{3})^5=\frac{4^5}{3^5}\)

 

Laws of Surds

 
Law 1 \(\sqrt{a} \sqrt{b}=\sqrt{ab}\)
Law 2 \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\)
Law 3 \(\sqrt{a^2}=a\)

 

Logarithms

\(2^3=8 \Leftarrow\Rightarrow log_2(8)=3\)

\(a^n=y \Leftarrow\Rightarrow log_a(y)=n\)

 
Law 1 \(log_a(xy)=log_a(x)+log_a(y)\) \(log_24+log_28=log_232\)
Law 2 \(log_a(\frac{x}{y})=log_ax-log_ay\) \(log_416-log_44=log_44\)
Law 3 \(log_ax^q=qlog_ax\) \(log_55^3=3log_55\)
Law 4 \(log_a1=0\) \(log_21=0\)
Law 5 \(log_a\frac{1}{x}=-log_ax\) \(log_3\frac{1}{27}=-log_327\)
Law 6 \(log_aa^x=x\) \(log_22^4=4\)
Law 7 \(a^{log_ax}=x\) \(10^{log_{10}100=100}\)
Law 8 \(log_bx=\frac{log_ax}{log_ab}\) \(log_416=\frac{log_216}{log_24}\)

 

Examples

 

\[ \displaylines{ \text{Solve }4^x=\frac{8}{\sqrt2}, x\in\mathbb{Q} \\\\ (2^2)^x=\frac{2^3}{2^{\frac{1}{2}}} \\\\ 2^{2x}=2^{3-\frac{1}{2}} \\\\ 2x=x\frac{1}{2} \\\\ x=\frac{2\frac{1}{2}}{2} \\\\ x=1\frac{1}{4} } \]

 

 

\[ \displaylines{ \text{Solve the equation: } \\\\ 3^{2x+1}-28(3x)+9=0 \\\\ (3^1)(3^{2x})-28(3x)+9=0 \\\\ 3(3^x)^2-28(3^x)+9=0 \\\\ \text{let }3^x=y \\\\ 3y^2-28y+9=0 \\\\ 3y^2-27y-1y+9=0 \\\\ 3y(y-9)-1(y-9)=0 \\\\ (3y-1)(y-9)=0 \\\\ 3y-1=0 \quad\text{ or }\quad y-9=0 \\\\ 3y=1 \quad\text{ or }\quad y=9 \\\\ y=\frac{1}{3} \quad\text{ or }\quad y=9 \\\\ 3^x=\frac{1}{3} \quad\text{ or }\quad 3^x=9 \\\\ \quad\quad\quad\quad\quad\quad\quad3^x=3^2 \\\\ \quad\quad\quad\quad\quad\quad\quad x=2 } \]

 

 

\[ \displaylines{ \text{Simplify:} \\\\ \sqrt{50}+\sqrt{8}+\sqrt{32} \\\\ \sqrt{25}\sqrt2+\sqrt4\sqrt2+\sqrt{16}\sqrt2 \\\\ 5\sqrt2+2\sqrt2+4\sqrt2 \\\\ 11\sqrt2 } \]

 

 

\[ \displaylines{ \text{Simplify: } \\\\ (\sqrt5+2\sqrt2)(\sqrt5-\sqrt2) \\\\ \sqrt5(\sqrt5-\sqrt2)+2\sqrt2(\sqrt5-\sqrt2) \\\\ 5-\sqrt{10}+2\sqrt{10}-2(2) \\\\ 5+\sqrt{10}-4 \\\\ 1+\sqrt{10} } \]

 

 

\[ \displaylines{ \text{Rationalise the Denominator: } \\\\ \text{i)}\quad \frac{5}{\sqrt5} \\\\ \frac{5}{\sqrt5}\cdot\frac{\sqrt5}{\sqrt5} \\\\ \frac{5\sqrt5}{5} \\\\ \sqrt5 \\\\ \\\\ \text{ii)}\quad \frac{4\sqrt{15}}{\sqrt{20}} \\\\ \frac{4\sqrt{15}}{\sqrt{20}}\cdot\frac{\sqrt{20}}{\sqrt{20}} \\\\ \frac{4\sqrt{300}}{20} \\\\ \frac{4\sqrt{100}\sqrt{3}}{20} \\\\ \frac{4(10)(\sqrt3)}{20} \\\\ \frac{40\sqrt2}{20} \\\\ 2\sqrt3 \\\\ \\\\ \text{iii)}\quad \frac{\sqrt2-\sqrt3}{\sqrt2+\sqrt3} \\\\ \frac{\sqrt2-\sqrt3}{\sqrt2+\sqrt3} \cdot \frac{\sqrt2-\sqrt3}{\sqrt2-\sqrt3} \\\\ \frac{\sqrt2(\sqrt2-\sqrt3)-\sqrt3(\sqrt2-\sqrt3)}{\sqrt2(\sqrt2-\sqrt3)+\sqrt3(\sqrt2-\sqrt3)} \\\\ \frac{2-\sqrt6-\sqrt6+3}{2-\sqrt6+\sqrt6-3} \\\\ \frac{5-2\sqrt6}{-1} \\\\ -5+2\sqrt6 } \]

 

 

\[ \displaylines{ \text{Evaluate: }\quad log_{16}64 \\\\ log_{16}64=x \\\\ 16^x=64 \\\\ (4^2)^x=4^3 \\\\ 2x=3 \\\\ x=\frac{3}{2} } \]

 

 

\[ \displaylines{ \text{Solve }\quad log_{10}(3x+1)=2 \\\\ 3x+1=10^2 \\\\ 3x+1=100 \\\\ 3x=100-1 \\\\ 3x=99 \\\\ x=33 } \]

 

 

\[ \displaylines{ \text{Solve }\quad log_2(x+3)=log_2(x-9)^2 \\\\ (x+3)=(x-9)^2 \\\\ x+3=x(x-9)-9(x-9) \\\\ x+3=x^2-18x+81 \\\\ x^2-19x+78=0 \\\\ x^2-13x-6x+78=0 \\\\ x(x-13)-6(x-13)=0 \\\\ (x-6)(x-13)=0 \\\\ x=6\quad\text{ or }\quad x=13 } \]

 

 

\[ \displaylines{ \text{Solve }\quad log_2(x+1)-log_2(x-1)=1,x\gt1 \\\\ log_2(\frac{x+1}{x-1})=1 \\\\ \frac{x+1}{x-1}=2^1 \\\\ \frac{x+1}{x-1}=2 \\\\ x+1=2(x-1) \\\\ x+1=2x-2 \\\\ 2x-x=2+1 \\\\ x=3 } \]

 

 

\[ \displaylines{ \text{Solve }\quad 4log_x2=log_2x+3\quad,x>1,x\in\mathbb{R} \\\\ log_x2^4=log_2x+3 \\\\ \frac{log_216}{log_2x}=log_2x+3 \\\\ \frac{4}{log_2x}=log_2x+3 \\\\ \text{Let }\quad log_2x=y \\\\ \frac{4}{y}=y+3 \\\\ 4=y^2+3y \\\\ y^2+3y-4=0 \\\\ (y-1)(y+4)=0 \\\\ y=1\quad\text{ or }\quad y=-4 \\\\ \\\\ log_2x=1\quad\text{ or }\quad log_2x=-4 \\\\ x=2\quad\quad\text{ or }\quad\quad x=\frac{1}{16} } \]