find $2*(4^x+2^x)=3^x-6^x+9$.

Question

I really don't know how to find x. I tried to diff but it won't works at all. Although I can find ans in wolframe but I want to know how to find it. Please help.

Answer 1

As said in comments, you need a numerical method and to consider that you are looking for the zero of function $$f(x)=2*(4^x+2^x)-(3^x-6^x+9)=2^{x+1}+2^{2 x+1}-3^x+6^x-9$$ Assuming that you prove that this is an increasing function, by inspection (or graphing), notice that $f(0)=-5$ and $f(1)=6$. So, there is a root somewhere between $0$ and $1$.

To get it, consider Newton method which, starting from a guess $x_0$, will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ where $$f'(x)=2^{x+1} \log (2)+2^{2 x+2} \log (2)-3^x \log (3)+6^x \log (6)$$

Let us use it using $x_0=\frac 12$. The iterates will be $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.50000000000000000000 \\ 1 & 0.64553363577073244675 \\ 2 & 0.63186649906103557062 \\ 3 & 0.63172424859838398232 \\ 4 & 0.63172423340791808566 \\ 5 & 0.63172423340791791247 \end{array} \right)$$ which is the solution for twenty signifiacnt figures.