42.What is the value of xin 5cdot ((1)/(4))^3x+4=40

To solve the equation \(5 \cdot \left(\frac{1}{4}\right)^{3x+4} = 40\), we can follow these steps:<br /><br />1. Divide both sides of the equation by 5:<br />\(\left(\frac{1}{4}\right)^{3x+4} = 8\)<br /><br />2. Take the logarithm of both sides of the equation:<br />\(\log\left(\left(\frac{1}{4}\right)^{3x+4}\right) = \log(8)\)<br /><br />3. Use the power rule of logarithms to simplify the left side:<br />\((3x+4) \cdot \log\left(\frac{1}{4}\right) = \log(8)\)<br /><br />4. Divide both sides of the equation by \(\log\left(\frac{1}{4}\right)\):<br />\(3x+4 = \frac{\log(8)}{\log\left(\frac{1}{4}\right)}\)<br /><br />5. Subtract 4 from both sides of the equation:<br />\(3x = \frac{\log(8)}{\log\left(\frac{1}{4}\right)} - 4\)<br /><br />6. Divide both sides of the equation by 3:<br />\(x = \frac{1}{3} \cdot \left(\frac{\log(8)}{\log\left(\frac{1}{4}\right)} - 4\right)\)<br /><br />Now, we can simplify the expression further:<br /><br />\(\log\left(\frac{1}{4}\right) = \log(4^{-1}) = -\log(4)\)<br /><br />\(\log(8) = \log(2^3) = 3\log(2)\)<br /><br />So, the expression becomes:<br /><br />\(x = \frac{1}{3} \cdot \left(\frac{3\log(2)}{-\log(4)} - 4\right)\)<br /><br />We can simplify the fraction inside the parentheses:<br /><br />\(\frac{3\log(2)}{-\log(4)} = \frac{3\log(2)}{-2\log(2)} = -\frac{3}{2}\)<br /><br />So, the expression becomes:<br /><br />\(x = \frac{1}{3} \cdot \left(-\frac{3}{2} - 4\right)\)<br /><br />Simplifying the expression inside the parentheses:<br /><br />\(-\frac{3}{2} - 4 = -\frac{3}{2} - \frac{8}{2} = -\frac{11}{2}\)<br /><br />So, the final expression for \(x\) is:<br /><br />\(x = \frac{1}{3} \cdot \left(-\frac{11}{2}\right) = -\frac{11}{6}\)<br /><br />Therefore, the value of \(x\) in the equation \(5 \cdot \left(\frac{1}{4}\right)^{3x+4} = 40\) is \(-\frac{11}{6}\).