Logarithmic Demand Functions: p = 5000 [ 1 - ( 4 / 4+e^(-0.002x) ) ]

[Jaime_Kartel]

Here is the full problem:

p = 5000 [1-(4/4+e^-0.002x)]

Where p is the price per unit and x is the number of units sold.
Find the number of units sold for prices of (a) p = 200

So far, my work is:

200 = 5000 [1-(4/4+e^-0.002x)]

Then divide the 5000 to equal:

.04 = 1-(4/4+e^-0.002x)

Now, I am somewhat lost on where to go from here. I assume I would subtract (1), then clear the negative on the right side which would leave me:

.96 = (4/4+e^-0.002x)

If this is so far correct, I am unsure what to do from here. I've tried various route, but none of them is the correct answer which is about 896 units.

Any help would be greatly appreciated. Thank you.

 

[ksdhart]

Well, the first thing I'd note is that you are missing some parentheses. What you've written is the following:

\(\displaystyle 0.96=\frac{4}{4}+e^{-0.002x}=1+e^{-0.002x}\)

I'm assuming, however, that you meant this, instead:

\(\displaystyle 0.96=\frac{4}{4+e^{-0.002x}}\)

In that case, you have a fraction on the right hand side, so let's do what we always do with fractions.

\(\displaystyle 0.96\cdot \left(4+e^{-0.002x}\right)=4\)

\(\displaystyle 3.84+0.96e^{-0.002x}=4\)

Now you continue from here. Keep working with it and simplifying and eventually you'll end up with e^-0.002x on one side. Then what do you think you would do to "clear" the e, and arrive at -0.002x = (something)?

 

[Jaime_Kartel]

Thank you! Yes you are correct, I did miss the parenthesis...

Can't believe I forgot that simple fraction issue...Greatly appreciated!