answer:To solve the equation, follow these steps: 1. **Convert to logarithmic form:** [ log(4^{2x + 3}) = log(1) ] 2. **Use the property of logarithms:** [ (2x + 3) log(4) = 0 ] 3. **Simplify and isolate the variable:** [ 2x log(4) + 3 log(4) = 0 ] [ 2x log(4) = -3 log(4) ] [ x = frac{-3 log(4)}{2 log(4)} ] [ x = -frac{3}{2} ] The solution is ( x = -frac{3}{2} ). **Practice exercises:** Solve for ( x ) in each of the following equations: a) ( 2^{3x - 7} = 5 ) b) ( 3^{2x + 3} = 4^{x - 5} ) c) ( 2^{4x - 9} = 3 cdot 3^{x + 6} ) Remember to keep your answers in exact form, unless instructed otherwise. Good luck!

answer:The Jacobian matrix of mathbf{r}(x, y, z) is given by: J(mathbf{r}(x, y, z)) = begin{bmatrix} frac{partial f}{partial x} & frac{partial f}{partial y} & frac{partial f}{partial z} frac{partial g}{partial x} & frac{partial g}{partial y} & frac{partial g}{partial z} frac{partial h}{partial x} & frac{partial h}{partial y} & frac{partial h}{partial z} end{bmatrix} Calculating each partial derivative, we get: frac{partial f}{partial x} = frac{1}{x + y^4}, quad frac{partial f}{partial y} = frac{4y^3}{x + y^4}, quad frac{partial f}{partial z} = 0 frac{partial g}{partial x} = 0, quad frac{partial g}{partial y} = frac{4y^3}{1 + y^8}, quad frac{partial g}{partial z} = 0 frac{partial h}{partial x} = 0, quad frac{partial h}{partial y} = 0, quad frac{partial h}{partial z} = e^z Therefore, the Jacobian matrix of mathbf{r}(x, y, z) is: J(mathbf{r}(x, y, z)) = begin{bmatrix} frac{1}{x + y^4} & frac{4y^3}{x + y^4} & 0 0 & frac{4y^3}{1 + y^8} & 0 0 & 0 & e^z end{bmatrix}

answer:To express the given product as a sum containing only sines or cosines of the expression {eq}y=sin (5 theta) sin (7 theta){/eq}, we can use the following formulas: {eq}2sin Asin B=cos(A-B)-cos(A+B) cos 2theta =cos^2theta -sin^2theta {/eq} First form: begin{align} 2y&=displaystyle 2sin (7 theta) sin (5 theta) [2ex] text {Using: } 2sin Asin B&=cos(A-B)-cos(A+B):[2ex] 2y&=displaystyle cos (2 theta)- cos (12 theta)[2ex] y&=boxed{displaystyle dfrac{1}{2} bigg(cos (2 theta)- cos (12 theta)bigg)} end{align} Second form: begin{align} y&=displaystyle dfrac{1}{2} bigg(cos (2 theta)- cos (12 theta)bigg) [2ex] text {Using: } cos 2theta &=cos^2theta -sin^2theta:[2ex] implies y&= boxed{dfrac 12bigg( cos^2theta-sin^2 theta- cos (12 theta) bigg)} end{align}

answer:The eigenvalues of the matrix are: lambda_1 = -1.478 - 9.273i, quad lambda_2 = -1.478 + 9.273i, quad lambda_3 = -1.043