what is the solution for this equation : $2^x-6(2)^{-x}=6$
I couldn't even make one step ! :(
I thought to use logarithms here but it wouldn't be useful .. it would make it more complicated
$\endgroup$3 Answers
$\begingroup$$$2^x-6(2)^{-x}=6$$ $$2^x-6/2^{x}=6$$ $$(2^x)^2-6\cdot2^{x}-6=0$$ $$2^x=\frac{6\pm\sqrt{60}}{2}$$ because $2^x>0$ $$2^x=\frac{6+\sqrt{60}}{2}=3+\sqrt{15}\iff x=\log_2{(3+\sqrt{15})}$$
$\endgroup$ $\begingroup$Substitute $y := 2^x$ and use $2^{-x} = 1/y$. Then solve for $y$ (how?) and solve the result for $x$.
$\endgroup$ $\begingroup$$2^x - \dfrac{6}{2^x} = 6$.
Let $t = 2^x$. This is now a simple equation:
$t - \dfrac{6}{t}=6$.
And an obligatory reminder to ignore any negative solutions, since you are working with real numbers.
$\endgroup$ 2
