qlogsqrt (x+1)-3logsqrt [3](x) 2x+(1)/(2)=10

Let's solve the given equations step by step.<br /><br />### Equation 1: \( q \log \sqrt{x+1} - 3 \log \sqrt[3]{x} \)<br /><br />First, let's simplify the logarithmic terms:<br /><br />1. \(\sqrt{x+1}\) can be written as \((x+1)^{1/2}\).<br />2. \(\sqrt[3]{x}\) can be written as \(x^{1/3}\).<br /><br />So the equation becomes:<br />\[ q \log (x+1)^{1/2} - 3 \log x^{1/3} \]<br /><br />Using the property of logarithms \(\log a^b = b \log a\), we get:<br />\[ q \cdot \frac{1}{2} \log (x+1) - 3 \cdot \frac{1}{3} \log x \]<br />\[ = \frac{q}{2} \log (x+1) - \log x \]<br /><br />This is the simplified form of the first equation.<br /><br />### Equation 2: \( 2x + \frac{1}{2} = 10 \)<br /><br />To solve for \(x\), follow these steps:<br /><br />1. Subtract \(\frac{1}{2}\) from both sides:<br /> \[ 2x = 10 - \frac{1}{2} \]<br /> \[ 2x = \frac{20}{2} - \frac{1}{2} \]<br /> \[ 2x = \frac{19}{2} \]<br /><br />2. Divide both sides by 2:<br /> \[ x = \frac{19}{4} \]<br /><br />So, the solution to the second equation is:<br />\[ x = \frac{19}{4} \]<br /><br />### Summary<br /><br />1. The simplified form of the first equation is:<br /> \[ \frac{q}{2} \log (x+1) - \log x \]<br /><br />2. The solution to the second equation is:<br /> \[ x = \frac{19}{4} \]