Find the value of each of the following, giving your answex 9y-:yb jvddt3l b5q;a/ucos7u+ l, f8qz7tnr as an integer.
1. ${\log }_{6}6$ .   
2. ${\log }_{6}9+{\log }_{6}4$ .   
3. ${\log }_{6}72-{\log }_{6}2$ .   

  Find the value of each of the following, giving your answer as an iwe)yuqw) opcm 73p )t/nteger.
1. ${\log }_{10}100$ .   
2. ${\log }_{10}50+{\log }_{10}2$ .   
3. ${\log }_{10}4-{\log }_{10}40$ .   

  Let p=ln 2 and q=ln 6 . Write down the following expressions in t0hkv k8bqdt6+bw b7mxy8qq;:rgz .d. erms of p and y87dqq.bbqwt 6 k0 ;rb dxh.g8+kvz: m q .
1. ln 12 =  ----待选择----p+q｜p-q｜2p+q 2q-p 
2. ln 3 =  ----待选择----p+q｜q-p｜2p+q 2q-p 
3. ln 48 =  ----待选择----2p+q｜3p-q｜3p+q 2q-p 

  Let ${\log }_{2}a=p,{\log }_{2}b=q,{\log }_{2}c=r$ . Write down the following expressions in terms of p, q and r .
1. ${\log }_2\left(\frac{ab}{c}\right)$ =  ----待选择----pqp+qp-qp/qp+q-r 
2. ${\log }_2\left(\frac{a^{2}c}{b^3}\right)$ =  ----待选择----2pq2p+3q2p+r-3qp-q-r 
3. ${\log }_{a}b$=  ----待选择----q/p2p+3q2p+r-3qp-q-r 

  $\text{ Consider }a={\log }_{63}64\times {\log }_{62}63\times {\log }_{61}62\times \cdots \times {\log }_{2}3\text{. Given that }a\in \mathbb{Z}\text{, find the value of }a\text{. }$   

  Let $a={\log }_{5}b$ , where b>0 . Write down each of the following expressions in terms of a .
1. ${\log }_5{b}^4$ =  ----待选择----3a4aa+14a 
2. ${\log }_5(25b)$ =  ----待选择----5aa+32+aa+1 
3. ${\log }_{25}b$ =  ----待选择----a/2a/4a/3a/5 

  1. Write the expression 38p(ee;ws/rqn h1 nb*i caav-$4\ln 2-\ln 8$ in the form of $\ln k$ , where $k\in \mathbb{Z}$ . lna a =   
2. Hence, or otherwise, solve $4\ln 2-\ln 8=-\ln (2x)$ .   

  $\text{ Solve the equation }{\log }_{3}x-{\log }_{3}5=1+{\log }_{3}4\text{ for }x\text{. }$   

  $\text{ Solve the equation }{\log }_{5}x-{\log }_{5}4=2+{\log }_{5}3\text{ for }x\text{. }$   

  $\text{ Solve the equation }{\log }_{5}x-{\log }_{5}4=2+{\log }_{5}3\text{ for }x\text{. }Giventhat{\log }_{a}2=5.$
1. Find the exact value of ${\log }_{a}32$ .   
2. Find the exact value of ${\log }_{\sqrt{a}}2$ .   
3 . Find the value of a , giving your answer correct to 3 significant figures. ≈   

  Given that ${\log }_{b}3=10$ .
1. Find the exact value of ${\log }_{b}81$ .   
2. Find the exact value of ${\log }_{b^2}3$ .   
3. Find the value of b , giving your answer correct to 3 significant figures.≈   

  $\text{ Find the values of }x\text{ when }{27}^{x+2}={\left(\frac{1}{9}\right)}^{2x+4}\text{. }$   

  1. Write down the value of bn,dl t0a +(ho
1. ${\log }_{2}8$ ;   
2. ${\log }_5\left(\frac{1}{25}\right)$   
3. ${\log }_{9}3$ .   
2. Hence solve ${\log }_{2}8+{\log }_5\left(\frac{1}{25}\right)+{\log }_{9}3={\log }_{16}x$ .   

  1. Write down the value oup picr dl2n2)afwntg*b ,sok 8u5+,5 f
1. ${\log }_{3}81$ ;   
2. ${\log }_2\left(\frac{1}{8}\right)$ ;   
3. ${\log }_{25}5$ .   
2. Hence solve ${\log }_{3}81+{\log }_2\left(\frac{1}{8}\right)+{\log }_{25}5={\log }_9$ x .   

  $\text{ Solve }{\log }_6(x)+{\log }_6(x-5)=2\text{, for }x>5\text{. }$   

  $\text{ Solve the equation }{\log }_2(x^2-2x+1)=1+{\log }_2(x-1)$   

  $\text{ Solve the equation }{\log }_3(x^2-4x+4)=1+{\log }_3(x-2)$   

  $\text{ Find the values of }x\text{ when }{25}^{x^2-2x}={\left(\frac{1}{125}\right)}^{4x+2}\text{. }$   

  Find the value of
1. ${\log }_{7}98-{\log }_{7}2$ ;   
2. ${49}^{{\log }_{7}6}$ .   

  1. ${\log }_{5}75-{\log }_{5}3$ ;   
2. ${25}^{{\log }_{5}8}$ .   

  $\text{ Solve the equation }{9}^x+2\cdot 3^{x+1}=1$ ${\log }_a(b+\sqrt{c})$ a =    b =    c =   

  Consider $f(x)={\log }_k(8x-2x^2)$ , for
$0<x<4$ , where k>0 .
The equation f(x)=3 has exactly one solution. Find the value of k .   

  Solve ${\log }_{\sqrt{3}}(\sin x)-{\log }_{\sqrt{3}}(\cos x)=1$, for $0<x<\frac{\pi }{2}$ . $\frac{\pi }{a}$ a =   

  $\text{ Solve the equation }{15}^{4a}={81}^{a+2}\text{ for }a\text{. Express your answer in terms of }\ln 3\text{ and }\ln 5\text{. }$   

  $\text{ Solve the equation }{14}^{6x}={64}^{x+3}\text{ for }x\text{. Express your answer in terms of }\ln 2\text{ and }\ln 7\text{. }$   -

  Find the integer values of vsr:f q8ywpafji4 -/ji:j. v7 a and b for which

$a+b{\log }_{4}7+60{\log }_{8}14=0$ .

a =    b =   

The first three terms of an infinite sequence, in order, )n. :izjsac 1:4bekhnare $2\ln x,q\ln x$,$\ln \sqrt{x}$ where x>0 .
First consider the case in which the series is geometric.
1. 1. Find the possible values of q .
2. Hence or otherwise, show that the series is convergent.
2. Given that q>0 and $S_{\mathrm{\infty }}=8\ln 3$ , find the value of x .

Now suppose that the series is arithmetic.
3. 1. Show that $q=\frac{5}{4}$ .
2. Write down the common difference in the form $m\ln x$ , where $m\in \mathbb{Q}$ .
4. Given that the sum of the first n terms of the sequence is $\ln \sqrt{x^5}$ , find the value of n .

  The first two terms of an infinite geokqvqc , -wu ze-7p+9co0n ee)bmetric sequence, in order, are

$3{\log }_{3}x,2{\log }_{3}x,\text{ where }x>0\text{. }$

1. Find the common ratio, r .   
2. Show that the sum of the infinite sequence is $9{\log }_3$ x .$a{\log }_{b}x$ a =    b =   

The first three terms of an arithmetic sequence, in order, are

${\log }_{3}x,{\log }_3\frac{x}{3},{\log }_3\frac{x}{9},\text{ where }x>0\text{. }$

3. Find the common difference d , giving your answer as an integer.

Let $S_6$ be the sum of the first 6 terms of the arithmetic sequence.   
4. Show that $S_6=6$ ${\log }_{3}x-15$ .$a{\log }_{b}x-c$ a =    b =    c =   
5. Given that $S_6$ is equal to one third of the sum of the infinite geometric sequence, find x , giving your answer in the form $a^p$ where a, $p\in \mathbb{Z}$ .$a^b$a =    b =   

  Solve the simultaneous equations: sbavyjix -4gh1 0zt5,

$\begin{array}{rl}1+2{\log }_{5}x& ={\log }_{5}7y\\ {\log }_7(6x-1)& =1+{\log }_{7}y\end{array}$

x =    y =    ; x =    y =   

  1. Show that ${\log }_{16}(\cos 2x+7)={\log }_4\sqrt{\cos 2x+7}$ .${\log }_a\sqrt{\cos bx+c}$ a =    b =    c =   
2. Hence, or otherwise, solve ${\log }_4(\sqrt{10}\cos x)={\log }_{16}(\cos 2x+7)$ , for $0<x<\frac{\pi }{2}$ .$\frac{\pi }{a}$ a =   

  1. Show that ${\log }_2\sqrt{3-\cos 2x}={\log }_4(3-\cos 2x)$ .${\log }_a(b-\cos cx)$ a =    b =    c =   
2. Hence, or otherwise, solve ${\log }_4(3\sin x)+\frac{1}{4}={\log }_2\sqrt{3-\cos 2x}$ , for $0<x<\frac{\pi }{2}$ .$\frac{\pi }{a}$ a =   

  The first three terms of a geohdid1s *xq.:gdtv;us6i6xl q m-q+:2;bk q gametric sequence are $\ln x^9,\ln x^3,\ln x$ , for x>0 .
1. Find the common ratio.   
2. Solve $\sum _{k=1}^{\mathrm{\infty }}{3}^{3-k}\ln x=27$ . $e^a$ a =