answer:Over the past two-hundred years, Western societies have become more secular, leading to a decline in the influence of religion on various aspects of culture, such as art, architecture, music, philosophy, politics, and cuisine.

answer:The given equation can be simplified by combining the logarithms with the same base: log (x-3) + log (-21x-11) = log (4x-19) Using the property log(a) + log(b) = log(ab), we can combine the logarithms: log((x-3)(-21x-11)) = log(4x-19) Now, the logarithms are equal, so the arguments must be equal: (x-3)(-21x-11) = 4x-19 Expanding the left side: -21x^2 - 11x + 63x + 33 = 4x - 19 Combine like terms: -21x^2 + 52x + 52 = 0 Divide by -1 to make the coefficient of x^2 positive: 21x^2 - 52x - 52 = 0 Factor the quadratic equation (if possible) or use the quadratic formula to find the roots: x = frac{-(-52) pm sqrt{(-52)^2 - 4 cdot 21 cdot (-52)}}{2 cdot 21} x = frac{52 pm sqrt{2704 + 4368}}{42} x = frac{52 pm sqrt{7072}}{42} x = frac{52 pm 4sqrt{441}}{42} x = frac{52 pm 4 cdot 21}{42} Now, we have two solutions: x_1 = frac{52 + 84}{42} = frac{136}{42} = frac{68}{21} x_2 = frac{52 - 84}{42} = frac{-32}{42} = -frac{16}{21} However, since -21x-11 and 4x-19 are within logarithms, x must be greater than 0 and 4x-19 must also be greater than 0. Thus, we discard x_2 = -frac{16}{21} as it is negative. The real solutions to the equation are: left{xto frac{68}{21}right}

answer:This function contains the inverse sine function. The derivative of the inverse sine function is defined as: {eq}displaystyle frac{d}{dx} sin^{-1} x = frac{1}{sqrt{1-x^2}} {/eq} Since we have a linear function inside the inverse sine function, we need to apply the Chain Rule. The derivative of a linear function is its coefficient. {eq}begin{align*} f'(x) &= frac{d}{dx} sin^{-1} left( frac{x}{2} right) &= frac{1}{sqrt{1-left( frac{x}{2} right)^2 }} cdot frac{d}{dx} left( frac{x}{2} right) &= frac{1}{sqrt{1-left( frac{x}{2} right)^2 }} cdot frac{1}{2} &= frac{1}{2sqrt{1-frac{x^2}{4}}} &= frac{1}{sqrt{4-x^2}} end{align*} {/eq} Therefore, the derivative of the function is {eq}f'(x) = frac{1}{sqrt{4-x^2}} {/eq}.

answer:The normalized vector is obtained by dividing each component by the vector's magnitude. The magnitude of the given vector is sqrt{left(frac{2}{3}right)^2 + left(frac{4}{3}right)^2 + left(-frac{4}{3}right)^2 + left(-frac{4}{3}right)^2 + 2^2} = sqrt{22}. Upon normalization, we get: left( begin{array}{c} frac{frac{2}{3}}{sqrt{22}} frac{frac{4}{3}}{sqrt{22}} frac{-frac{4}{3}}{sqrt{22}} frac{-frac{4}{3}}{sqrt{22}} frac{2}{sqrt{22}} end{array} right) = left( begin{array}{c} frac{1}{sqrt{22}} sqrt{frac{2}{11}} -sqrt{frac{2}{11}} -sqrt{frac{2}{11}} frac{1}{sqrt{11}} end{array} right)