Question

Use the formula

2 = (1+ r/100)ⁿᵗ

to find the doubling time t, in years, for an investment at r% compounded n times per year. write this exponential statement in logarithmic form.

Answer 1

This computation provides an approximate doubling time of 25.46 years when using the change-of-base formula for logarithms.

The formula given is 2=1+ r/n^{nt}), where:

r is the annual interest rate (in decimal form),

n is the number of times the interest is compounded per year, and

t is the time in years.

For an investment at 11% compounded quarterly (n=4), we have:

2= 1 + (0.11/4)^4t

First, solve for t using logarithmic form to solve for the exponent:

2= (1+ 0.11/4)^4t

2=(1.0275)^4t

Taking the natural logarithm of both sides:

ln(2)=ln(1.0275)^4t

ln(2)=4t⋅ln(1.0275)

Now, solve for t:

t= ln(2)/ 4⋅ln(1.0275)

​Using a calculator to compute this:

t≈ ln(2)/ 4⋅ln(1.0275)

≈ 0.6931/ 4⋅0.0272

≈6.42

So, the doubling time for an investment at 11% compounded quarterly is approximately 6.42 years.

If you're required to use a change-of-base formula (logarithmic form), the change-of-base formula for logarithms is:

`log_b(a)= (log a)/(log b)`

​You could use this formula to find the doubling time:

t= log(2)/ log(1.0275)

t≈ 0.6931/ 0.0272

≈25.46

This computation provides an approximate doubling time of 25.46 years when using the change-of-base formula for logarithms.

Question

Use the formula 2 = 1 + r 100n nt to find the doubling time t, in years, for an investment at r % compounded n times per year. Write this exponential statement in logarithmic form for an investment at 11% compounded quarterly. t = Use a change-of-base formula to find the doubling time (in years) for this investment. (Round your answer to two decimal places.) t = Incorrect: Your answer is incorrect.