Exponential and logarithms: Practice Questions

Solve the equation:
\(\log_2 x + \log_2 3 = \log_2 12\)

Find the set of values of \(x\) for which:
\(2 \log_3 x - \log_3 (x - 2) = 2\)

Solve the exponential equation \(4^{x+1} - 9 \cdot 2^x + 2 = 0\).

The variables \(x\) and \(y\) satisfy the equation \(y = k a^x\), where \(k\) and \(a\) are positive constants. A graph of \(\log_{10} y\) against \(x\) is a straight line passing through the points \((0, 0.5)\) and \((4, 2.1)\).
Find the values of \(k\) and \(a\), giving your answers to 2 decimal places.

Given that \( \log_b 2 = x \) and \( \log_b 3 = y \), express \( \log_b \left( \frac{8}{9} \right) \) in terms of \( x \) and \( y \).

Solve the equation \( 5^{2x+1} = 12 \), giving your answer to three decimal places.

The variables \( y \) and \( x \) satisfy the relationship \( y = Ax^n \). A straight line is produced by plotting \( \log_{10} y \) against \( \log_{10} x \). Given the line passes through \( (0, 2) \) and \( (3, 8) \), find the values of the constants \( A \) and \( n \).

(a) Use the laws of logarithms to write 2 log_a x - log_a 4 as a single logarithm in terms of x and a.
(b) Given that 2 log₂ x - log₂ 4 = 3, find the exact value of x.

(a) Solve the equation log₂(x + 3) + log₂(x + 1) = 3.
(b) Solve the equation 3^2y - 12(3^y) + 27 = 0.