Question

find all solutions (x) to each of the following equations. separate multiple answers with commas. use no solution or none if there are no solutions.
(3e^{10x}=5e^{8x}) has solutions (x =)
(7e^{10x}=4e^{10x}+5e^{8x}) has solutions (x =)
(7e^{10x}=4e^{10x}+20e^{8x}) has solutions (x =)
(\frac{e^{10x}}{e^{10x}+5e^{8x}}=\frac{4}{7}) has solutions (x =)
note: you can earn partial credit on this problem.

Explanation:

Step1: Solve $3e^{10x}=5e^{8x}$

Divide both sides by $e^{8x}$: $3e^{10x - 8x}=5$, so $3e^{2x}=5$. Then $e^{2x}=\frac{5}{3}$. Take natural - log of both sides: $2x=\ln(\frac{5}{3})$, and $x = \frac{1}{2}\ln(\frac{5}{3})$.

Step2: Solve $7e^{10x}=4e^{10x}+5e^{8x}$

Subtract $4e^{10x}$ from both sides: $3e^{10x}=5e^{8x}$. This is the same as the first equation, so $x=\frac{1}{2}\ln(\frac{5}{3})$.

Step3: Solve $7e^{10x}=4e^{10x}+20e^{8x}$

Subtract $4e^{10x}$ from both sides: $3e^{10x}=20e^{8x}$. Divide both sides by $e^{8x}$: $3e^{2x}=20$. Then $e^{2x}=\frac{20}{3}$. Take natural - log of both sides: $2x=\ln(\frac{20}{3})$, and $x=\frac{1}{2}\ln(\frac{20}{3})$.

Step4: Solve $\frac{e^{10x}}{e^{10x}+5e^{8x}}=\frac{4}{7}$

Cross - multiply: $7e^{10x}=4(e^{10x}+5e^{8x})$. Expand: $7e^{10x}=4e^{10x}+20e^{8x}$. Subtract $4e^{10x}$ from both sides: $3e^{10x}=20e^{8x}$. Divide both sides by $e^{8x}$: $3e^{2x}=20$. Then $e^{2x}=\frac{20}{3}$. Take natural - log of both sides: $2x=\ln(\frac{20}{3})$, and $x=\frac{1}{2}\ln(\frac{20}{3})$.

Answer:

$\frac{1}{2}\ln(\frac{5}{3}),\frac{1}{2}\ln(\frac{5}{3}),\frac{1}{2}\ln(\frac{20}{3}),\frac{1}{2}\ln(\frac{20}{3})$