2. Solve for x in log (2 x+4)-4 operatorname(oy)(x-1) =3 log 2 (3mks)

To solve for \( x \) in the equation \( \log (2x+4) - 4 \log (x-1) = 3 \log 2 \), we can follow these steps:<br /><br />1. Combine the logarithmic terms on the left side of the equation:<br /> \[ \log (2x+4) - \log (x-1)^4 = 3 \log 2 \]<br /><br />2. Use the properties of logarithms to simplify the equation:<br /> \[ \log \left( \frac{2x+4}{(x-1)^4} \right) = 3 \log 2 \]<br /><br />3. Rewrite the equation in exponential form:<br /> \[ \frac{2x+4}{(x-1)^4} = 2^3 \]<br /><br />4. Simplify the equation:<br /> \[ \frac{2x+4}{(x-1)^4} = 8 \]<br /><br />5. Cross-multiply to get rid of the fraction:<br /> \[ 2x+4 = 8(x-1)^4 \]<br /><br />6. Expand and simplify the equation:<br /> \[ 2x+4 = 8(x^4 - 4x^3 + 6x^2 - 4x + 1) \]<br /> \[ 2x+4 = 8x^4 - 32x^3 + 48x^2 - 32x + 8 \]<br /><br />7. Move all terms to one side of the equation:<br /> \[ 8x^4 - 32x^3 + 48x^2 - 32x + 8 - 2x - 4 = 0 \]<br /> \[ 8x^4 - 32x^3 + 48x^2 - 34x + 4 = 0 \]<br /><br />8. Divide the entire equation by 2 to simplify:<br /> \[ 4x^4 - 16x^3 + 24x^2 - 17x + 2 = 0 \]<br /><br />9. Solve the quadratic equation using a suitable method, such as factoring, completing the square, or using the quadratic formula.<br /><br />After solving the quadratic equation, we can find the values of \( x \) that satisfy the original equation.