# Natural logs test problem

1. Sep 4, 2006

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,

2. Sep 4, 2006

### J77

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

3. Sep 4, 2006

### danago

I got stuck on the same question :(

heres the working i did for it after the test.

4. Sep 4, 2006

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

5. Sep 4, 2006

### Staff: Mentor

eax * e-ax = eax-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

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

$$1.5x = ln 2$$

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

Last edited: Sep 4, 2006
6. Sep 4, 2006

### J77

I just didn't want to give the game away