if $1750 is invested at the end of each year in an annuity that pays 5% compounded annually, the number of…

if $1750 is invested at the end of each year in an annuity that pays 5% compounded annually, the number of years it takes for the future value to amount to $70,000 is given by $t = \\log_{1.05}3$. use the change of base formula to find the number of years until the future value is $70,000$. it will take approximately $\\square$ years for the future value to amount to $70,000$. (simplify your answer. round up to the nearest whole number.)

if $1750 is invested at the end of each year in an annuity that pays 5% compounded annually, the number of years it takes for the future value to amount to $70,000 is given by $t = \\log_{1.05}3$. use the change of base formula to find the number of years until the future value is $70,000$. it will take approximately $\\square$ years for the future value to amount to $70,000$. (simplify your answer. round up to the nearest whole number.)

Answer

Explanation:

Step1: Apply the change - of - base formula

The change - of - base formula for a logarithm (\log_{a}b=\frac{\ln b}{\ln a}) (or (\frac{\log b}{\log a})). Here, (a = 1.05) and (b = 3), so (t=\log_{1.05}3=\frac{\ln 3}{\ln 1.05}) (we can also use common logarithms: (t = \frac{\log 3}{\log 1.05})). Using a calculator, (\ln 3\approx1.0986) and (\ln 1.05\approx0.0488).

Step2: Calculate the value of (t)

[t=\frac{\ln 3}{\ln 1.05}\approx\frac{1.0986}{0.0488}\approx22.51]

Answer:

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