The given equations are:

1) sin2x - cosx = 2sinx - 1
2) log3x + 4log9x = 9

Let's solve each equation separately:

1) sin2x - cosx = 2sinx - 1

Using the trigonometric identity sin(2x) = 2sinx*cosx, we can rewrite the equation as:

2sinx*cosx - cosx = 2sinx - 1

cosx(2sinx - 1) = 2sinx - 1

cosx = 1

However, the range of cosine function is [-1, 1]. Therefore, there are no solutions to this equation.

2) log3x + 4log9x = 9

Using the property of logarithms that states logb(x) + logb(y) = logb(xy), we can combine the logarithms:

log3x + log(9x)^4 = 9

log(3x * (9x)^4) = 9

log(3x 9^4 x^4) = 9

log(3 9^4 x^5) = 9

log(3 6561 x^5) = 9

log(19683x^5) = 9

19683x^5 = 10^9

x^5 = 10^9 / 19683

x^5 = 50625

x = 50625^(1/5)

x ≈ 4.764

Therefore, the solution to the second equation is x ≈ 4.764.