Question

Log4 (2^2x-√3 cos x-sin2x)=x

Answer 1

Ответ:

π/2+πn, n∈z

(-1)^k·π/3+πk, k∈z

Пошаговое объяснение:

по определению логарифма запишем

2^2х-√3cosx-sin2x=4^x

2^2x=4^x сделав подстановку, перенесем слагаемое 4^x c противоположным знаком в правую часть, получим

-√3cosx-sin2x=0

-√3cosx-2sinx·cosx=0

-cosx(√3+2sinx)=0

cosx=0          √3+2sinx=0

Решим каждое из двух уравнений.

cosx=0

x=π/2+πn, n∈z

√3+2sinx=0

2sinx=-√3

sinx=-√3/2

По общей формуле найдем корни

x=(-1)^k·arcsin(-√3/2)+πk, k∈z

x=(-1)^(k-1)·π/3 +πk, k∈z

Answer 2

It seems like you have an equation involving logarithms and trigonometric functions:

$\log_4(2^{2x} - \sqrt{3} \cos(x) - \sin^2(x)) = x$

Solving this equation analytically might not be straightforward, as it involves both logarithms and trigonometric functions. However, we can outline the general steps you might take to solve it:

1. Simplify the equation: Try to simplify the expression within the logarithm as much as possible.

2. Use properties of logarithms: If possible, use properties of logarithms to isolate the logarithm on one side of the equation.

3. Apply trigonometric identities: Use trigonometric identities to simplify the trigonometric terms in the equation.

4. Attempt to isolate $x$: Once the equation is simplified, try to isolate $x$ on one side of the equation. This might involve using properties of logarithms, exponentials, and trigonometric identities.

5. Numerical methods: If the equation cannot be solved analytically, you might resort to numerical methods such as iteration or graphing to approximate the solution.

Due to the complexity of the equation, the solution might not have a simple closed-form expression, and numerical methods might be necessary. If you're looking for a specific solution, I recommend using computational tools or software that can handle equations involving both logarithmic and trigonometric functions.