serious issues with some problems.

1. The ABC corporation issues a bond that pays 10,000$ after 10 years. If the initial investment is 5,000$ what is the interest rate per year, compounded annually.

The equation to use is A = P ( 1 + r )^t

Where A = Ending amount

P = Starting amount

r = interest rate

t = time

Right now I have this

10,000 = 5,000 ( 1 + r ) ^ 10

2 = ( 1 + r ) ^ 10

$\displaystyle

Log_((1+r)2=10$

But i'm not sure where to go from here. Any ideas?

2. Mary wants to invest money in a 7-year saving certificate with an interest rate of 6% per year, compounded continuously. How much will she need to invest to have $1000 at the end of the term.

This uses the equation A = Pe^rt Where all variables have the same value is before, and of course e is e (2.7.........)

So far I have

1000=Pe^(7 x .06)

1000=Pe^.42

$\displaystyle ln(1000)/ln(.42)=lnP$

And now I have no idea what to do

3. Write the estimate in exponential form (Using the definition of a logarithm)

$\displaystyle ln10 = 2.303$

With this one I have absolutely no idea what I should do.

4. A water filter is constructed from a foam that REMOVES 25% of impurities for each inch of foam. How many inches of foam are needed for the filter to remove 50% of the impurities.

I know that this can be modeled with $\displaystyle A = Pa^x$ and also with A = Pe^(rx)

P = concentration of impurities before filtration

A = concentration of impurities in filtered water

x = thickness of foam

I understand that $\displaystyle .5P = Pa^x $

But I can't move much further from there.

Please don't just give answers. I really need to understand how to solve each problem. Thank you in advance.