Solve for $x$ without using a calculator.

Question

How can I solve for $x$ $$3e^{-2x}-6e^{-x}=72$$

If someone can go step by step and do the problem, it would be really appreciable.

Answer 1

Substitute $e^{-x}=t$.

Now, Note that since $e^{-2x}=\left( e^{-x} \right)^2$, we have that $$3e^{-2x}-6e^{-x}-72=0 \iff 3t^2-6t-72=0 \iff 3(t+4)(t-6)=0$$ From factorization. Since $t>0$, we have that $e^{-x}=t=6$.

So $-x=\ln 6$, or $x=-\ln 6$.

Answer 2

Hint: substitute $e^{-x} = t$ to obtain a quadratic equation in t.