How to Solve Exponential Equations with Natural Logs?

[pavadrin]

hey
2day i had a test in intro calc and was given the following problem to solve:

$$50e^(2x) = 100e^(0.5x)$$

i was quite annoyed at this problem because i didn’t know how to solve it. although i do know that natural logs need to be taken on both sides to solve it, however the coefficient of the e really had me confused. any help would be greatly appreciated, thanks,
Pavadrin

 

[J77]

What happens if you multiply both sides by exp(-0.5x)... ?

 

[danago]

I got stuck on the same question :(

heres the working i did for it after the test.

http://img145.imageshack.us/img145/3503/logszp0.gif

 

[pavadrin]

if both sides are multiplied by exp(-0.5x) would that make it:

$$50e^2.5x = 100e^x$$

but i don't see how that helps, thanks for the relpy though

 

[Astronuc]

e^ax * e^-ax = e^ax-ax = e⁰ = 1

danago is correct.

Using ln,

$$50e^{2x} = 100e^{0.5x}$$

start by dividing the equation by 50

$$e^{2x} = 2e^{0.5x}$$

take natural log

$$2x = ln 2\,+\,0.5x$$

$$1.5x = ln 2$$

$$x = \frac{ln 2}{1.5}$$

 

[J77]

danago is correct.

I just didn't want to give the game away