How to solve $e^{5k}-e^{-5k}=3$?

Question

$$e^{5k}-e^{-5k}=3$$ How do I solve for k using the substitution $y=e^{5k}$ by making it into a quadratic equation?

Answer 1

Hint: recall that $e^{-5x}=\frac{1}{e^{5x}}\equiv \frac1y$ where the last equivalence is using your substituion.

From here you should know how to go about.

Super Hint: Using the substitution together with the first hint we obtain $$y-\frac1y=3$$ and multiplying with $y$ we obtain $$y^2-1=3y$$ Now you definitely need to solve this one yourself (and keep in mind that $y=0$ cannot be a solution, although in this case it is anyway not a candidate)

Answer 2

$$e^{5k}-e^{-5k}=3\Longleftrightarrow (e^{5k})^2- 3e^{-5k} -1 = 0\Longleftrightarrow X^2 -3X -1 =0 $$ With $X=e^{5k}$, $$\Delta = 13$$

Then $$X=\frac{3 \pm\sqrt{13}}{2}$$

But $e^{5k} =X>0$ and $\frac{3 -\sqrt{13}}{2}<0$ so $$X=\frac{3 +\sqrt{13}}{2}$$

hence $$k = \frac15 \ln\left( \frac{3 +\sqrt{13}}{2}\right)$$