Chapter 1

$$ \frac{\log _{3} 81}{\log _{3} 9}\left(36^{1-\log _{6} 2}+49^{-\log _{7} 6}\right) $$

$$ \frac{\log _{\sqrt{2}} 16}{\log _{4} \sqrt{2}}\left[\log _{\sqrt{2}}(2 \cdot \sqrt[4]{2})+100^{\frac{1}{2} \log 8-2 \log 2}\right] $$

$$ 10^{\frac{1}{2} \log 9-\log 5+\log 2} \cdot 7^{\log _{3 \sqrt{3}} 27} $$

$$ \log _{\sqrt{6}} 3 \cdot \log _{3} 36+\log _{\sqrt{3}} 8 \cdot \log _{4} 81 $$

$$ 72 \log _{2}\left(\sqrt{\frac{1}{5}}\right) \cdot \log _{25} \sqrt[3]{2}+10 \log _{2}\left(\frac{\sqrt[5]{8}}{2}\right) $$

$$ 3^{\frac{2}{5} \log _{3} 32-\frac{1}{3} \log _{3} 64+\log _{3} 10} . $$

$$ (0.2)^{\frac{1}{2}\left(9 \log _{0.2} 2-3 \log _{0.2} 4\right)} $$

$$ (\sqrt{2})^{3 \log _{\sqrt{2}} 5-2 \log _{\sqrt{2}} 25-\log _{\sqrt{2}} 10+2 \log _{\sqrt{2}} \sqrt{5}} $$

$$ (\log 2+\log 5+\log 300-\log 3) \cdot 3^{\frac{1}{5 \log _{5} 3}} $$

$$ \left(\log _{8} 27-\log _{0.5} \frac{1}{3}\right) \cdot\left(\frac{\log _{3} 12}{\log _{36} 3}-\frac{\log _{3} 4}{\log _{108} 3}\right) $$

$$ \frac{\log _{2} \sqrt[3]{\frac{2}{3}}}{\log _{2}^{2} \sqrt{7}}-\frac{2 \log _{\sqrt{7}} \sqrt[3]{\frac{2}{3}}}{\log _{\sqrt{3}} \sqrt{7}}-\log _{\sqrt[6] 7} \sqrt[3]{\frac{2}{3}} \cdot \log _{\sqrt{7}} \sqrt[3]{\frac{2}{3}} $$

$$ 2^{\log _{5} 3} \cdot\left(\frac{1}{3}\right)^{1-\log _{5} 2.5} \cdot \log _{9} 2 \cdot \log _{4} 81 $$

$$ \log _{3} 2 \cdot \log _{4} 3 \cdot \log _{5} 4 \cdot \log _{6} 5 \cdot \log _{7} 6 \cdot \log _{8} 7 $$

$$ \text { Given that } \log _{6} 2=a, \text { find } \log _{24} 72 \text { in terms of } a \text { . } $$

$$ \text { Given that } \log _{36} 8=a, \text { find } \log _{36} 9 \text { in terms of } a \text { . } $$

$$ \text { Given that } \log _{4} 125=a \text { , find } \log 64 \text { in terms of } a \text { . } $$

$$ \text { Given that } \log _{100} 3=a \text { and } \log _{100} 2=b, \text { find } \log _{5} 6 \text { in terms of } a \text { and } b \text { . } $$

$$ \text { Given that } \log _{6} 15=a \text { and } \log _{12} 18=b, \text { find } \log _{25} 24 \text { in terms of } a \text { and } b \text { . } $$

$$ \text { If } \ln 2 \cdot \log _{a} 625=\log 16 \cdot \ln 10, \text { then find the value of } a \text { . } $$

$$ \text { Prove that } \log _{b} a \log _{c} b \log _{a} c=1 $$

$$ \text { Prove that } \log _{b} a \log _{c} b \log _{d} c \log _{a} d=1 $$

$$ \text { If } \log _{a}(a b)=x, \text { then evaluate } \log _{b}(a b) \text { in terms of } x \text { . } $$

$$ \text { Prove that } \frac{\log _{a} n}{\log _{a b} n}=1+\log _{a} b $$

$$ \text { Prove that } \log _{a b} x=\frac{\log _{a} x \log _{b} x}{\log _{a} x+\log _{b} x} $$

$$ \text { If } a^{2}+b^{2}=7 a b, \text { prove that } \log \frac{1}{3}(a+b)=\frac{1}{2}[\log a+\log b] \text { . } $$

$$ \text { Show that } \frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\cdots \cdots+\frac{1}{\log _{43} n}=\frac{1}{\log _{43 !} n} \text { . } $$

$$ \text { If } n=1983 ! \text { , compute the sum } \frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\frac{1}{\log _{4} n}+\cdots \cdots+\frac{1}{\log _{1983} n} \text { . } $$

$$ \text { If } y=a^{\frac{1}{\left(1-\log _{a} x\right)}} \text { and } z=a^{\frac{1}{\left(1-\log _{a} y\right)}}, \text { prove that } x=a^{\frac{1}{1-\log _{a} z}} \text { . } $$

$$ \text { If } \frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}, \text { prove that } a^{a} \cdot b^{b} \cdot c^{c}=1 \text { . } $$

$$ \text { If } \frac{\log x}{q-r}=\frac{\log y}{r-p}=\frac{\log z}{p-q}, \text { prove that } x^{q+r} \cdot y^{r+p} \cdot z^{p+q}=x^{p} \cdot y^{q} \cdot z^{r} . $$

$$ \text { Prove that } \log _{a} n \log _{b} n+\log _{b} n \log _{c} n+\log _{c} n \log _{a} n=\frac{\log _{a} n \log _{b} n \log _{c} n}{\log _{a b c} n} \text { . } $$

$$ \text { If } a>0, c>0, b=\sqrt{a c}, a \neq 1, c \neq 1, a c \neq 1 \text { and } n>0, \text { prove that } \frac{\log _{a} n}{\log _{c} n}=\frac{\log _{a} n-\log _{b} n}{\log _{b} n-\log _{c} n} \text { . } $$

$$ \text { Prove that if } x=\log _{c} b+\log _{b} c, y=\log _{a} c+\log _{c} a, z=\log _{b} a+\log _{a} b \text { then } x y z=x^{2}+y^{2}+z^{2}-4 $$

$$ \begin{aligned} &\text { If } f(x)=a x^{2}+b x+c, \text { find } f(0), f(1), f(-1), f(a) \text { and } f(b) . \text { Ans. } c, a+b+c, a-b+c,\\\ &\left.a^{3}+a b+c, a b^{2}+b^{2}+c\right\\} \end{aligned} $$

$$ \text { If } \left.f(x)=\frac{x-1}{x} \text { and } g(x)=x^{2}+1, \text { find } g(f(1)) \text { and } f(g(-1)) \text { . \\{ Ans. } 1, \frac{1}{2}\right\\} $$

$$ \begin{aligned} &\text { If } f(x)=x^{2}+3 x+1 \text { and } g(x)=2 x-3 \text { . Find fog, gof, fof and gog. \\{Ans. }(f \circ g)(x)=4 x^{2}-6 x+1 \text { ; }\\\ &\left.(g o f)(x)=2 x^{2}+6 x-1 ;(f o f)(x)=x^{4}+6 x^{3}+14 x^{2}+15 x+5 ;(g o g)(x)=4 x-9\right\\} \end{aligned} $$

$$ \text { Let } f(x)=x^{2} \text { and } g(x)=2 x+1 \text { . Find fog and gof. Also show that } f o g \neq \text { gof } \text { . } $$

$$ \text { If } f(x)=x^{2}+2 \text { and } g(x)=\frac{x}{x-1} . \text { Find fog and gof. } $$

$$ \text { If } f(x)=x^{2}-3 x+2, \text { find } f(f(x)) $$

$$ \text { Find } \phi(\psi(x)) \text { and } \psi(\phi(x)) \text { if } \phi(x)=x^{2}+1 \text { and } \psi(x)=3^{x} \text { . } $$

$$ \text { If } f(x)=\sin x \text { and } g(x)=x^{2}, \text { then find } f \circ g(x) \text { and } \operatorname{gof}(x) \text { . } $$

$$ \text { If } f(x)=x \cos x \text { and } g(x)=\frac{x}{1+x^{2}} \text { , then find } f \circ g(x) \text { and } g o f(x) $$

$$ \text { If } f(x)=x^{2}-\frac{1}{x^{2}}, \text { prove that } f(x)=-f\left(\frac{1}{x}\right) \text { . } $$

$$ \text { If } f(x)=x+\frac{1}{x}, \text { prove that }(f(x))^{3}=f\left(x^{3}\right)+3 f\left(\frac{1}{x}\right) \text { . } $$

$$ \text { Given the function } f(x)=\frac{a^{x}+a^{-x}}{2},(a>0), \text { show that } f(x+y)+f(x-y)=2 f(x) f(y) \text { . } $$

Given the function \(\begin{aligned} f(x) &=3^{-x}-1, &-1 \leq x<0 \\ &=\tan \frac{x}{2}, & & 0 \leq x<\pi \\ &=\frac{x}{x^{2}-2}, & & \pi \leq x \leq 6 . \end{aligned}\) Find:- i. \(f(-1) ;\\{\) Ans. 2\(\\}\) ii. \(f\left(\frac{\pi}{2}\right) ;\\{\) Ans. 1\(\\}\) iii. \(f\left(\frac{2 \pi}{3}\right) ;\\{\) Ans. \(\sqrt{3}\\}\) iv. \(f(4) ;\left\\{\right.\) Ans. \(\left.\frac{2}{7}\right\\}\) v. \(f(6) .\left\\{\right.\) Ans. \(\left.\frac{3}{17}\right\\}\)

Let \(f(x)=1\), if \(x\) is a rational number \(=0\), if \(x\) is a irrational number. Find:- i. \(f\left(\frac{1}{3}\right) ;\\{\) Ans. 1\(\\}\) ii. \(f(\sqrt{7}) ;\\{\) Ans. 0\(\\}\) iii. \(f\left(\frac{22}{7}\right) ;\\{\) Ans. 1\(\\}\) iv. \(f(\pi) ;\\{\) Ans. 0\(\\}\) v. \(f(e) ;\\{\) Ans. 0\(\\}\) vi. \(f(f(1.4327)) ;\\{\) Ans. 1\(\\}\) \text { vii. } f(f(\sqrt{3})) \text { . }

Given \(f(x)=x^{2}, \quad x \geq 0\) \(=x, \quad x<0\) and \(\begin{aligned} g(x) &=\frac{1}{x}, & & x \geq 1 \\ &=1, & & x<1 . \end{aligned}\) Determine the following functions:- i. \(h(x)=x \times g(x)\); ii. \(\phi(x)=f(x)+g(x)\) iii. \(\psi(x)=f(x) \times g(x)\) \\{Ans. i. \(h(x)=1, \quad x \geq 1\) ii. \(\quad \phi(x)=x^{2}+\frac{1}{x}, \quad x \geq 1\) \(=x^{2}+1, \quad 0 \leq x<1\) \(=x+1, \quad x<0 ;\) iii. \(\quad \psi(x)=x, \quad x \geq 1\) \(=x^{2}, \quad 0 \leq x<1\) \(=x, \quad x<0\\}\)

$$ \text { Determine the function } f(x)=|x| \times \operatorname{sgn} x .\\{\text { Ans. } f(x)=x\\} $$

Determine the following functions:- i. \(\quad f_{1}(x)=\sin |x|\); ii. \(\quad f_{2}(x)=\sin ^{-1}(\operatorname{sgn} x)\); iii. \(f_{3}(x)=|x|^{2}\); iv. \(\quad f_{4}(x)=(\operatorname{sgn} x)^{2}\); v. \(f_{5}(x)=e^{\operatorname{sgn} x}\). \\{Ans. i. \(f_{1}(x)=\sin x, \quad x \geq 0\) \(=-\sin x, \quad x<0\) ii. \(f_{2}(x)=\frac{\pi}{2}, \quad x>0\) \(=0, \quad x=0\) \(=-\frac{\pi}{2}, \quad x<0\) iii. \(f_{3}(x)=x^{2}\) iv. \(f_{4}(x)=1, \quad x \neq 0\) \(=0, \quad x=0\) v. \(f_{5}(x)=e, \quad x>0\) \(=1, \quad x=0\) $$ \left.=\frac{1}{e}, \quad x<0\right\\} $$