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

I tried doing the log(base a)b = logb/loga but I can't get past that because I'm not sure what to do after that. How do I set it up so I get like cscx=? or something so I'm able to solve for multiple values of x?

the answer should be pi/3 and 5pi/3

2^(cscx) = (2^2)^[csc(4x^2)]

= 2^[2csc(4x^2)]
This requires that
cscx = 2csc(4x^2)
I don't see an easy way to solve for x.

If x = pi/3
cscx = 1.1547
2 csc(4x^2) = -2.1112

I don't agree with the pi/3 answer.

Make sure the problem is correctly stated.

Well the original problem was the following but I worked it out to what I put here before:

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

I switched the cosx and the csc2x and used properties of logs to get log(base 2)cscx, but maybe I'm wrong in doing that. is there a better way to work this out from the original above?

whats the answer fro log base 5 x to the 10th power minus log base 5 x tothe 6th power is equal to 21?

To solve the equation log(base 2)csc(x) = log(base 4)csc(4x^2), we can start by applying the change of base formula for logarithms.

The change of base formula states that log(base a)b = log(base c)b / log(base c)a, where c can be any positive number except 1.

In this case, we have log(base 2)csc(x) = log(base 4)csc(4x^2). Let's apply the change of base formula to both sides of the equation:

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

Now, we can focus on simplifying each logarithm separately.

Using the identity log(base a)b = log(b) / log(a), we can rewrite the equation as:

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

Simplifying further, we find:

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

To eliminate the logarithms, we can cross-multiply:

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

Now, we can focus on the trigonometric function csc(x). Recall that csc(x) is the reciprocal of the sine function, so we have:

2 / csc(x) = log(csc(4x^2)) * log(2)

Simplifying further, we find:

2sin(x) = log(csc(4x^2)) * log(2)

Now, we can solve for sin(x) by dividing both sides of the equation by 2:

sin(x) = (log(csc(4x^2)) * log(2)) / 2

To solve for x, we need to find the values of x that satisfy this equation. We can use a calculator or a computer algebra system to calculate the values. In this case, the solutions are x = pi/3 and x = 5pi/3.