Question

Expressy as a function of x. The constant C is a positive number.

$3\ln y=\frac{1}{2}\ln \left(2x+1\right)-\frac{1}{3}\ln \left(x+4\right)+\ln C$

Short answer

$y=\sqrt[3]{\frac{C{\left(2x+1\right)}^{\frac{1}{2}}}{{\left(x+4\right)}^{\frac{1}{3}}}}$

Step-by-step solution

Step 1. Given information.

Consider the given information.

$3\ln y=\frac{1}{2}\ln \left(2x+1\right)-\frac{1}{3}\ln \left(x+4\right)+\ln C$

Step 2. Calculation.

Use the logarithm property to solve.

$\begin{array}{l}3\ln y=\frac{1}{2}\ln \left(2x+1\right)-\frac{1}{3}\ln \left(x+4\right)+\ln C\\ \ln y^3=\ln {\left(2x+1\right)}^{\frac{1}{2}}-\ln {\left(x+4\right)}^{\frac{1}{3}}+\ln C\\ \ln y^3=\ln \frac{{\left(2x+1\right)}^{\frac{1}{2}}}{{\left(x+4\right)}^{\frac{1}{3}}}+\ln C\\ \ln y^3=\ln \frac{C{\left(2x+1\right)}^{\frac{1}{2}}}{{\left(x+4\right)}^{\frac{1}{3}}}\\ y^3=\frac{C{\left(2x+1\right)}^{\frac{1}{2}}}{{\left(x+4\right)}^{\frac{1}{3}}}\\ y=\sqrt[3]{\frac{C{\left(2x+1\right)}^{\frac{1}{2}}}{{\left(x+4\right)}^{\frac{1}{3}}}}\end{array}$