Logarithmic and exponential functions：練習問題

Using the laws of logarithms, simplify the expression \( 2\log_a 6 - \log_a 9 \).

Solve the equation \( 3^{2x} - 4(3^x) + 3 = 0 \) for \( x \).

A relationship between \( y \) and \( x \) is given by \( y = ab^x \). A graph of \( \ln y \) plotted against \( x \) results in a straight line with a gradient of \( 0.5 \) and a \( y \)-intercept of \( 2 \).
Find the value of \( y \) when \( x = 4 \).

The variables \( x \) and \( y \) satisfy the relationship \( y = Ae^{kx} \). When \( \ln y \) is plotted against \( x \), a straight line is obtained passing through the points \( (0, \ln 2) \) and \( (2, \ln 18) \). Find the value of \( k \).

Solve the equation \(\ln(2x - 3) = 2\ln 3 - \ln(x - 1)\) for \(x\), giving your answer to 3 significant figures.

Solve the inequality \(5^{2x-1} > 3^{x+2}\), giving your answer in terms of natural logarithms.