log(base 2)cotx - 2log(base 4)csc2x = log(base 2)cosx

the answer should be x = pi/3, 5pi/3

not sure how to work it out though, any help? I tried canceling the bases but that led to a dead end.

Think about log2 and log4. If N is a number, log2(N)=1/2 log4(N)

I am uncertain if the second term is csc^2 x or csc 2x. But my hint above should lead you to the solution.

It's csc 2x, and I got it down to this before but I don't know if I can use your hint from there or if I should start from square one:

log(base 2)csc(x) = log(base 4)csc(4x^2)

Would that become the following?

log(base 2)csc(x) = 2log(base2)csc(4x^2)

no, of course not.

did you mean this?
2log2 cscx= log4 cscx
That is correct.

To solve the given equation, we can start by simplifying each term using logarithm properties. Let's break it down step by step:

1. Start with the equation:
log(base 2)(cot(x)) - 2 * log(base 4)(csc^2(x)) = log(base 2)(cos(x))

2. Apply the log property: log(base b)(a^m) = m * log(base b)(a), to simplify each term:
log(base 2)(cot(x)) - log(base 4)(csc^2(x))^2 = log(base 2)(cos(x))

3. Simplify the term with cotangent (cot(x)):
log(base 2)(cot(x)) - log(base 4)(1 / sin^2(x))^2 = log(base 2)(cos(x))

4. Apply the log property: log(base b)(1/a) = -log(base b)(a), to further simplify:
log(base 2)(cot(x)) + 2 * log(base 4)(sin^2(x)) = log(base 2)(cos(x))

5. Rewrite log(base 4)(sin^2(x)) using a base conversion formula:
log(base 2)(cot(x)) + 2 * (log(base 2)(sin^2(x)) / log(base 2)(4)) = log(base 2)(cos(x))

6. Simplify log(base 2)(4) = 2, and bring the denominator up:
log(base 2)(cot(x)) + 2 * (log(base 2)(sin^2(x)) / 2) = log(base 2)(cos(x))

7. Simplify further:
log(base 2)(cot(x)) + log(base 2)(sin^2(x)) = log(base 2)(cos(x))

8. Combine the terms on the left side using the logarithm property: log(base b)(a) + log(base b)(c) = log(base b)(a * c):
log(base 2)(cot(x) * sin^2(x)) = log(base 2)(cos(x))

9. Set the expressions inside the logarithms equal to each other since they have the same base:
cot(x) * sin^2(x) = cos(x)

10. Rewrite cot(x) as cos(x) / sin(x):
(cos(x) / sin(x)) * sin^2(x) = cos(x)

11. Simplify:
cos(x) * sin(x) = cos(x)

12. Divide both sides by cos(x):
sin(x) = 1

13. Since sin(x) = 1 has multiple solutions, we need to find the values of x that satisfy it. The solutions occur when x is equal to π/2 plus any even multiple of π:
x = π/2 + 2nπ, where n is an integer.

14. However, we need to check if the solution x = π/2 satisfies the equation we obtained earlier:
cot(π/2) * sin^2(π/2) = cos(π/2)
cot(π/2) * 1^2 = 0
Since cot(π/2) is not defined, x = π/2 is not a valid solution.

15. Now, let's find the other values of x that satisfy sin(x) = 1:
x = π/2 + 2nπ, where n is an odd integer.

16. Substitute odd values for n into x = π/2 + 2nπ to find x:
x = π/2 + 2(1)π = π/2 + 2π = 5π/2
x = π/2 + 2(3)π = π/2 + 6π = 13π/2
...

17. Simplify the solutions by reducing them to the interval [0, 2π]:
x = π/2, 5π/2, 13π/2, 21π/2, ...

18. Among the solutions obtained, select the values of x that were in the original interval [0, 2π]:
x = π/2, 5π/2

Therefore, the solutions to the equation are x = π/2 and x = 5π/2.