How do I solve this analytically $3^x=9x$ One of my friends ask me how to solve this equation analytically $3^x=9x$. Looking at it I guess 3 is the answer and I also plot a graph of line $9x$ and the curve $3^x$, they intersect at 3.

But, what I want is to give an analytical solution of the equation. I started 

$3^x=9x$

$3^{x-2}=x$

$(x-2)\ln3=\ln x$

How can I continue?
$$3^x=9x \Rightarrow 1 =\frac {9x}{3^x} \Rightarrow 1= 9x \cdot e ^{-x\ln 3}\Rightarrow \frac{1}{9}=x \cdot e ^{-x\ln 3} \Rightarrow$$

$$\Rightarrow \frac{-\ln 3}{9}=(-x \ln 3)\cdot e^{-x\ln 3}\Rightarrow W\left(\frac{-\ln 3}{9}\right)=-x \ln 3\Rightarrow$$

$$x= \frac{-W\left(\frac{-\ln 3}{9}\right)}{\ln 3}$$

where W is Lambert W function .
I am going to give three methods of solving this problem and they are:
$1$. Differentiation -                                                                               It can be broken down into a simpler form by finding  the derivative of both sides.
$\frac{d(3^x)}{dx} = \frac{d(9x)}{dx}$
$x(3^x - 1) = 9$
$x = 9$
$3^{x - 1}=9$
$3^{x - 1} = 3^2$
Equating the exponents,
$x - 1 = 2$
$x = 3$
Thus $x = 3 \ or \ 9$.
But since $9$ does not satisfy the equation.
$x = 3$.
$2$. Graphical approach -
This can be done by plotting a graph of $y=9x$ and $y=3^x$ and the $x$ component of the intersection is going to be $3$.
Thus $x=3$.
3. Value analyzing - This done by replacing the variable with figures starting from zero to see which one satisfies the equation and you will find out the equation is true when $x$ is $3$.
Thus $x=3$.