Exponentials and logarithms: แบบฝึกหัด

Find the exact solutions to the equation \( 5^{2x} - 12(5^x) + 35 = 0 \).

Solve the equation \( \log_2 x + \log_4 x = 6 \) for \( x \).

Solve the equation \(\log_2(x) = 5\) for \(x\).

Solve the equation \( 3^{2x-1} = 10 \), giving your answer for \( x \) to 3 decimal places.

Using the laws of logarithms, simplify the expression \(\log_a(2x^3) + \log_a(\frac{1}{x})\) into a single logarithm in its simplest form.

Using the laws of logarithms, solve the following equations:
(a) \(\log_{2}(x+3) + \log_{2}(x-3) = 4\), giving your answer as a simplified surd if necessary.
(b) \(2\log_{3}(y+1) - \log_{3}(2y-1) = 1\).

Solve the equation \(3^{2x} - 10(3^x) + 9 = 0\).