1. ## Exponential Functions

Once the initial publicity surrounding the release of a new book is over, sales of the hardcover edition tend to decrease exponentially. At the time publicity was discontinued, a certain book was experiencing sales of 25,000 copies per month. One month later, sales of the book had dropped to 10,000 copies per month. What will the sales be after 1 more month?
I have no idea on how to solve this problem. Please show me a step by step solution! The answer is 4000.

And also need assistance with the following question...

Find $\displaystyle f(2)$ if $\displaystyle f(x) = e^{kx}$ and $\displaystyle f(1) = 20$.

Here's what I did...

$\displaystyle 20 = e^{k \times 1}$
$\displaystyle e^{20} = e^{k \times 1}$
$\displaystyle 20 = e$

$\displaystyle f(2) = e^{20 \times 2}$
$\displaystyle f(2) = 2.3539$ x $\displaystyle 10^{17}$

2. For your second question: $\displaystyle 20=f(1)=e^k\rightarrow k=\log 20$ so $\displaystyle f(2)=e^{2\cdot \log 20}=\left(e^{\log 20}\right)^2=20^2=\boxed{400}$.

3. Originally Posted by Macleef
I have no idea on how to solve this problem. Please show me a step by step solution! The answer is 4000.

And also need assistance with the following question...

Find $\displaystyle f(2)$ if $\displaystyle f(x) = e^{kx}$ and $\displaystyle f(1) = 20$.

Here's what I did...

$\displaystyle 20 = e^{k \times 1}$
$\displaystyle e^{20} = e^{k \times 1}$
$\displaystyle 20 = e$

$\displaystyle f(2) = e^{20 \times 2}$
$\displaystyle f(2) = 2.3539$ x $\displaystyle 10^{17}$

Step 1 to Step 2 is wrong. How have you turned 20 into $\displaystyle e^{20}$?

You should have

$\displaystyle 20 = e^k$

$\displaystyle k = \ln{20}$.

Therefore $\displaystyle f(x) = e^{x\ln{20}} = e^{\ln{20^x}} = 20^x$.

Thus $\displaystyle f(2) = 20^2 = 400$.

4. Originally Posted by Macleef
$\displaystyle 20 = e^{k \times 1}$
$\displaystyle e^{20} = e^{k \times 1}$
$\displaystyle 20 = e$

$\displaystyle f(2) = e^{20 \times 2}$
$\displaystyle f(2) = 2.3539$ x $\displaystyle 10^{17}$

How did you get that?? If you take a look at the first two lines you can see that it just can be true. (It would mean that $\displaystyle 20=e^{20}$..) I suppose that you meant $\displaystyle 20 = k$ in the third line, then it would at least come from the line above and the other two lines would come from that...