Exponents & Logs

[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. $\log_{25} 5$. [3]
      

2. Hence solve $\log_3 81 + \log_2\Big(\dfrac{1}{8}\Big) + \log_{25} 5 = \log_{9} x$.[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. $\log_7 98 - \log_7 2$; [2]
   

2. $49^{\log_7 6}$. [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

[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. [3]
   

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