Exponentials and logarithms: แบบฝึกหัด

Evaluate \(\log_3 27 + \log_2 16\).

Solve the equation \(\log_4(2x + 1) = 2\).

Find the set of values of \( x \) that satisfy the logarithmic equation:
\( \log_2 x + 6 \log_x 2 = 5 \)

Solve the exponential equation \ \(3^{2x-1} = 27\)\.

Solve the equation \(\log_2(x^2 - 6x) = 3 + \log_2(x - 6)\) for \(x > 6\).

Find the exact values of \( x \) for which \( \log_3 x - 6\log_x 3 = 1 \).

Solve the simultaneous equations:
(i) \( \log_{2}x + \log_{2}y = \log_{2}18 \)
(ii) \( \log_{3}x - \log_{3}y = 2 \)

Give your answers in the form \( k\sqrt{2} \), where \( k \) is an integer or a fraction.