Question 606717: Please help me solve this:
How long does it take for an initial investment of $7,000 earning 5.58% interest, compounded continuously , to double? (First find the exact solution and then round to two decimal places.)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Where  is the future value,  is the present value, i.e. the initial investment,  is the base of the natural logarithms which is the limit of ^n) as  becomes large,  is the annual interest rate expressed as a decimal, and  is the number of years.
Regardless of the amount of the initial investment, in order for it to double,  and therefore  must equal 2.
Hence:
and solve for
Take the natural log of both sides. Actually, the log of any base will work and, given correct arithmetic, will provide the correct answer. However, since the equation involves , the arithmetic is greatly simplified by using the natural log.
\ =\ \ln(2))
Use
\ =\ n\,\cdot\,\log_b(a))
To write
\ =\ \ln(2))
Then use
\ =\ 1)
to write
)
and finally
}{r})
Just use your calculator to divide ) by the interest rate. Don't forget that the interest rate must be expressed as a decimal.
After that, toss the following into your financial formulas toolkit:
An initial investment of at an interest rate of compounded continuously will increase by a factor of  in }{r}) years.
John
My calculator said it, I believe it, that settles it
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