Logarithmic functions and indices: แบบฝึกหัด

Simplify the expression \(\frac{\sqrt{18} + \sqrt{32}}{\sqrt{2}}\).

Given that \(\log_3 x - 2\log_x 3 = 1\), find the possible values of \(x\).

The equation \(\log_x 6561 + 2 = \log_3 x\) has two real solutions, \(\alpha\) and \(\beta\).
Find the value of the product \(\alpha\beta\).

Solve the equation \(8^{x-1} = 16\) for \(x\).

Given that \(\log_{3} (x+2) - \log_{3} x = 2\), find the value of \(x\).

Solve the equation \(\log_5(2x + 7) - \log_5(x - 1) = 1\).

Solve the equation \( 25^{x-1} = 125^{x+2} \).

Solve the equation \( \log_3 (x+5) - \log_3 (x-3) = 2 \).

Solve the equation \( 27^{2x-1} = \frac{1}{9^x} \).

Solve the equation \( 2 \log_3 x = \log_3 (x+6) \).