Question

Solve ln(3x+4)−3ln(3)=ln(2x+1). Round answers to nearest hundredth.

Answer 1

$\ln (3x+4)-3\ln (3)=\ln (2x+1)$

We can operate on that expression as shown below

$\begin{gathered} \ln (3x+4)-3\ln (3)=\ln (2x+1) \\ \Rightarrow\ln (3x+4)-\ln (3^3)=\ln (2x+1);\ln (z^y)=y\cdot\ln (z) \\ \Rightarrow\ln (\frac{3x-4}{3^3})=\ln (2x+1);\ln (\frac{z}{y})=\ln (z)-\ln (y) \end{gathered}$

Remember that the function 'ln' is injective.This means that,

$\ln (z)=\ln (y)\Rightarrow z=y;y,z\in(0,\infty)$

So,

$\begin{gathered} \ln (\frac{3x-4}{3^3})=\ln (2x+1) \\ \Rightarrow\frac{3x-4}{3^3}=2x+1 \\ \Rightarrow\frac{3x-4}{27}=2x+1 \end{gathered}$

And this is simply a usual equation with one unknown. Solving for x,

$\begin{gathered} \frac{3x-4}{27}=2x+1 \\ \Rightarrow2x-\frac{1}{9}x=\frac{4}{27}-1 \\ \Rightarrow\frac{17}{9}x=-\frac{23}{27} \\ \Rightarrow x=-\frac{23}{51} \end{gathered}$

Now, we need to round to the nearest hundredth

$\begin{gathered} x=-\frac{23}{51}=-0.4209\ldots \\ x\approx-0.45 \end{gathered}$

Thus, the answer is x=-0.45 once we have rounded it