\[Maximum mark: 5\]

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





\[Maximum mark: 5\]

Solve the equation $\log_2(x^2-2x+1) = 1 + \log_2(x-1)$.





\[Maximum mark: 6\]



1.  Write down the value of 
    1.  $\log_3 81$;

    2.  $\log_2\Big(\dfrac{1}{8}\Big)$; 
    3.  $\log_{25} 5$. :marks[3]

     

2.  Hence solve
    $\log_3 81 + \log_2\Big(\dfrac{1}{8}\Big) + \log_{25} 5 = \log_{9} x$.:marks[3]




\[Maximum mark: 5\]

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





\[Maximum mark: 6\]

Find the value of 



1.  $\log_7 98 - \log_7 2$; :marks[2]

2.  $49^{\log_7 6}$. :marks[4]






\[Maximum mark: 5\]

Solve the equation $9^x + 2\cdot3^{x+1} = 1$.





\[Maximum mark: 5\]

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





\[Maximum mark: 5\]

Find the integer values of $a$ and $b$ for which 
$$
\begin{aligned}
a + b\log_4 7 + 60\log_{8} 14 &= 0.\end{aligned}
$$






\[Maximum mark: 8\]

The first three terms of a geometric sequence are $\ln x^9$, $\ln x^3$,
$\ln x$, for $x > 0$. 



1.  Find the common ratio. :marks[3]

2.  Solve $\displaystyle \sum_{k=1}^\infty 3^{3-k}\ln x = 27$.
    :marks[5]




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