Question

-Exponential and Logarithmic functions-Solve. Express the answer exactly using natural logs.

Answer 1

We want to find the solutions for the following equation

`4^(3x+1)=9^(2x)`

Using the following property of the natural log

`\ln a^b=b\ln a`

We can rewrite our expression applying the natural log on both sides of the equation.

`\begin{gathered} 4^(3x+1)=9^(2x) \\ \ln 4^(3x+1)=\ln 9^(2x) \\ (3x+1)\ln 4=2x\ln 9 \end{gathered}`

Using the distributive property

`\begin{gathered} (3x+1)\ln 4=2x\ln 9 \\ 3x\ln 4+\ln 4=2x\ln 9 \\ 3x\ln 4-2x\ln 9=-\ln 4 \\ x(3\ln 4-2\ln 9)=-\ln 4 \\ x=-(\ln4)/(3\ln4-2\ln9) \\ x=(\ln4)/(2\ln9-3\ln4) \end{gathered}`