Logarithmic Functions: Mastery Test

Type the correct answer in the box. Round your answer to the hundredth.

An investment in a savings account grows to three times the initial value after [tex]t[/tex] years. If the rate of interest is 5%, compounded continuously, [tex]t = [/tex] years.

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Answer :

To determine how many years it will take for an investment to grow to three times its initial value with an interest rate of 5% compounded continuously, we can use the formula for continuous compounding:

[tex]\[ A = P \cdot e^{(rt)} \][/tex]

Where:
- \( A \) is the final amount of the investment,
- \( P \) is the initial amount of the investment,
- \( r \) is the annual interest rate,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.

Given that the final amount \( A \) is three times the initial amount \( P \), we can write the equation as:

[tex]\[ 3P = P \cdot e^{(0.05t)} \][/tex]

To simplify, divide both sides by \( P \) (assuming \( P \neq 0 \)):

[tex]\[ 3 = e^{(0.05t)} \][/tex]

Next, to solve for \( t \), take the natural logarithm (ln) of both sides of the equation:

[tex]\[ \ln(3) = \ln(e^{(0.05t)}) \][/tex]

Using the property of logarithms that \( \ln(e^x) = x \), this simplifies to:

[tex]\[ \ln(3) = 0.05t \][/tex]

Now, isolate \( t \) by dividing both sides by 0.05:

[tex]\[ t = \frac{\ln(3)}{0.05} \][/tex]

Using a calculator to find the natural logarithm of 3:

[tex]\[ \ln(3) \approx 1.0986122886681098 \][/tex]

So:

[tex]\[ t = \frac{1.0986122886681098}{0.05} \approx 21.972245773362193 \][/tex]

Rounding this result to the nearest hundredth, we get:

[tex]\[ t \approx 21.97 \][/tex]

Thus, it will take approximately 21.97 years for the investment to grow to three times its initial value with a continuous compounded annual interest rate of 5%.