Using either logarithms or a graphing calculator, find the...

Using either logarithms or a graphing calculator, find the time required for each initial amount to be at least equal to the final amount.
$\$ 5000,$ deposited at 4$\%$ compounded quarterly, to reach at least $\$ 9000$

Key Concepts

Compound Interest

This concept relates to the process by which an investment’s earnings, from either interest or capital gains, are reinvested to generate additional earnings over time. It is typically modeled by an exponential equation where the amount grows as a function of the initial principal, the interest rate, and the number of compounding periods, illustrating how the investment accumulates value over time.

Exponential Growth

Exponential growth describes the process where a quantity increases at a rate proportional to its current value. In the context of finance, this is seen when interest is compounded, leading to increasingly larger amounts of accrued interest over time, which is critical for understanding how investments appreciate in value.

Logarithms

Logarithms are the inverse operations of exponentiation. They are used to solve for variables in exponent equations, such as determining the time it takes for an investment to reach a certain value when it is growing exponentially. This is done by taking the logarithm of both sides of the compound interest equation to isolate the time variable.

Compounding Frequency

This concept refers to the number of times per year that interest is calculated and added to the investment. It significantly affects the growth of the investment, as more frequent compounding results in higher returns over the same period. It is an essential factor in the compound interest formula and impacts the overall exponential growth of the invested amount.

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