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Interest rate is the percentage at which your investment grows per period of time. It's crucial to understand that this rate greatly affects the amount of money you will accumulate over time. In this exercise, the annual interest rate is given as 8%. This means if you invest \(1000, at the end of the year, you would earn 8% of \)1000 which is $80. However, interest is compounded quarterly in this case. This means your interest is calculated and added to your investment four times a year. To find the quarterly rate, we divide the annual rate by 4 (since interest is compounded quarterly). So, the quarterly interest rate is \(\frac{8%}{4} = 2% = 0.02\). Each quarter, you earn 2% on your investment.

Investment growth refers to how much your initial investment increases over time. In compound interest scenarios, growth doesn't just depend on the initial amount, but also on the interest gained. Each period (quarter, year, etc.), interest is added not just to the initial sum but to the accumulated amount up to that point. This compound effect allows your money to grow exponentially rather than linearly. In our exercise, with a quarterly compound interest rate, the value of the investment after t years is described by the formula: \[ A(t) = 1000 \times \bigg(1 + 0.02\bigg)^{4t} \] This illustrates that the investment's growth accelerates over time due to compounding.

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are key when dealing with compound interest because they model the accelerated growth nature of compounding. In our example, the formula \(A(t) = 1000 \times (1.02)^{4t}\) demonstrates an exponential function. Here, \(1.02\) is the base (slightly larger than 1 due to the 2% interest each quarter), and \(4t\) is the exponent, representing the number of compounding periods. Exponential functions have the distinctive property of growing faster as time passes, which is why the value of an investment under compound interest increases significantly over longer periods.

Logarithms are the inverse operations of exponentiation. They are used to solve problems where the exponent (time in these problems) is the unknown. To find the time required for an investment to double, we use logarithms. Starting with the formula for compounded growth: \(2000 = 1000 \times (1.02)^{4t}\), we first simplify it to \(2 = (1.02)^{4t}\). Applying the natural logarithm to both sides, we get \(\text{ln}(2) = \text{ln}\big((1.02)^{4t}\big)\). Using the rule \(\text{ln}(a^b) = b \times \text{ln}(a)\), it becomes \(\text{ln}(2) = 4t \times \text{ln}(1.02)\). Finally, solving for \(t\), we find \(t = \frac{\text{ln}(2)}{4 \times \text{ln}(1.02)} \), revealing the time needed for the investment to double.