Find the value of each of the following, :ib ijiwg(f;b 0: 5qsdgiving your answer 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 anss.:bfdf r ih.-wer as an integer.
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 exp72.jykagd k dv:pt*3s ressions in terms of p 7ap*:d yd.ks vg2jk3tand 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. $\log _{2}\left(\frac{a b}{c}\right)$ =  ----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}(25 b) $ =  ----待选择----5aa+32+aa+1 
3. $\log _{25} b$ =  ----待选择----a/2a/4a/3a/5 

  1. Write the expressio 2 7srexhgwclum0h44 3n $4 \ln 2-\ln 8$ in the form of $\ln k$ , where $k \in \mathbb{Z}$ . lna a =   
2. Hence, or otherwise, solve $4 \ln 2-\ln 8=-\ln (2 x)$ .   

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

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

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

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

  1. Write down the valu1d aw:yn+s;)p jvqa 5ze of
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 oj( vk,uz-cyq4i +oe fxplm5 lpt /+epd27::ewf
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 .   

  $\text { Solve } \log _{6}(x)+\log _{6}(x-5)=2 \text {, for } x>5 \text {. }$   

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

  $\text { Solve the equation } \log _{3}\left(x^{2}-4 x+4\right)=1+\log _{3}(x-2)$   

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

  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 $f(x)=\log _{k}\left(8 x-2 x^{2}\right)$ , for
$0\lt x \lt4$ , 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 $0\lt x \lt \frac{\pi}{2}$ . $\frac{\pi}{a}$ a =   

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

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

  Find the integer values of a and b for io4;byy4q;*h)8yhxn whj)h bwhich

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

a =    b =   

The first three terms of an infinite sequence, in order, are w::lfw: voak) $2 \ln x, q \ln x$,$ \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 $S_{\infty}=8 \ln 3$ , 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 geometric sequence, in ot5pi)z)kie( arder, 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 uh6r nv8-7a 2hbburq+ equations:

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

x =    y =    ; x =    y =   

  1. Show that $\log _{16}(\cos 2 x+7)=\log _{4} \sqrt{\cos 2 x+7}$ .$\log _{a} \sqrt{\cos b x+c}$ a =    b =    c =   
2. Hence, or otherwise, solve $\log _{4}(\sqrt{10} \cos x)=\log _{16}(\cos 2 x+7)$ , for $0\lt x\lt \frac{\pi}{2}$ .$\frac{\pi}{a}$ a =   

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

  The first three terms of a geometric s smr5/y+)rvxsequence are $\ln x^{9}, \ln x^{3}, \ln x$ , for x>0 .
1. Find the common ratio.   
2. Solve $\sum_{k=1}^{\infty} 3^{3-k} \ln x=27$ . $e^a$ a =