Exponents & Logs

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

1. $\log_{10} 100$. [2]
   

2. $\log_{10} 50 + \log_{10} 2$. [2]
   

3. $\log_{10} 4 - \log_{10} 40$. [3]
   

[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: 6]

1. Write the expression $4\ln 2 - \ln 8$ in the form of $\ln k$, where $k \in \mathbb{Z}$. [3]
   

2. Hence, or otherwise, solve $4\ln 2 - \ln 8 = -\ln (2x)$. [3]
   

[Maximum mark: 5]

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

[Maximum mark: 6]

Find the value of

1. $\log_5 75 - \log_5 3$; [2]
   

2. $25^{\log_5 8}$. [4]
   

[Maximum mark: 5]

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

[Maximum mark: 7]

Solve the simultaneous equations: