Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am having trouble solving the equation

$$3e^{−x+2} = 5e^{x-1}$$

Any help would be appreciated. Thanks.

share|cite|improve this question
Hint: take the natural logarithm of both sides! – Nigel Overmars Jan 6 at 15:38
hint: Try making the problem unnecessarily difficult by turning it into a quadratic equation. – John Joy Jan 7 at 16:41
@JohnJoy um... I think you should either take out the part "unnecessarily difficult" or remove that comment altogether and make it an answer. – Simple Art Jan 8 at 22:10
@SImple Art needs to lighten up a little bit. – John Joy Jan 9 at 0:43
@JohnJoy LoL, ok. – Simple Art Jan 9 at 0:58


Take log in both sides to find $$\ln3+(-x+2)=\ln5+x-1$$

share|cite|improve this answer
@user303134, See – lab bhattacharjee Jan 7 at 5:04

$$3e^{−x+2} = 5e^{x-1}$$ Taking the natural $\log$ of both sides gives: $$\ln (3e^{−x+2}) = \ln(5e^{x-1})$$ $$\ln3+(2-x) = \ln5+(x-1) ~~~(\text{ since } \ln(ab)=\ln a + \ln b).$$ So $$\ln 3 - \ln 5+3=2x.$$ But $\ln\left(\frac{a}{b}\right)=\ln a -\ln b$. $$\therefore x=\frac{1}{2}\Big(3+\ln\left(\frac{3}{5}\right)\Big).$$

share|cite|improve this answer

we have $\frac{3}{5}=e^{2x-3}$ by the power rule and then $\ln(\frac{3}{5})=2x-3$ can you proceed?

share|cite|improve this answer

You can also make the equation much simpler just by remembering your properties of exponents: $$3e^{−x+2} = 5e^{x-1}$$ $$3e^{−x}e^{2} = 5e^{x}e^{-1}$$ multiply across by e $$3e^{-x}e^{3}=5e^{x}$$ multiply across by $e^{x}$ and divide by $5$, take the natural log of both sides... $$3e^{3} = 5e^{2x}$$ Now you've got the $x$ on one side, :) rest ain't so bad $$\frac{3}{5}e^{3} = e^{2x}$$ $$\ln(\frac{3}{5}e^{3}) = \ln(e^{2x})$$ $$\ln(\frac{3}{5}e^{3}) = 2x$$ $$\therefore x = \frac{1}{2}\ln(\frac{3}{5}e^{3})$$

share|cite|improve this answer

$$3e^{-x+2} = 5e^{x-1}$$ $$\frac{3e^2}{e^x} = \frac{5e^x}{e^1}$$ $$\dots = \dots$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.