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

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