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Exponential equations are mathematical expressions that describe situations where a quantity grows or decays at a rate proportional to its current value. This is the essence of exponential growth, or in the reverse, exponential decay.
In the example given in the original exercise, the formula for exponential growth function is used to predict the future balance of a savings account. The formula is as follows:
\[A(t) = A_0 \times (1 + r)^t\]
where:

• \(A(t)\) is the amount of money at time \(t\).
• \(A_0\) is the initial deposit.
• \(r\) is the interest rate (expressed as a decimal).
• \(t\) is the time in years.

To solve the exponential equation, you often isolate the term with the exponent \(t\), allowing you to find the 'time' it takes for the initial amount to grow to a certain value. In many cases, solving for \(t\) requires taking a logarithm of both sides, as was done in the step-by-step solution provided in the textbook.

The natural logarithm is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. It's denoted as \(\ln\) and is a fundamental concept in mathematics, especially in the fields of calculus and complex analysis.
In the context of solving exponential equations, such as calculating how long it will take for money to increase from an initial deposit to a final amount in a savings account, the natural logarithm becomes a powerful tool. For instance, to isolate the exponent in the equation:
\[\frac{A(t)}{A_0} = (1 + r)^t\]
you take the natural logarithm of both sides. This results in:
\[\ln\left(\frac{A(t)}{A_0}\right) = \ln((1 + r)^t)\]
By exploiting the properties of logarithms, namely that \(\ln(a^b) = b \ln(a)\), this becomes:
\[t \cdot \ln(1 + r) = \ln\left(\frac{A(t)}{A_0}\right)\]
which can be rearranged to solve for \(t\). This step can sometimes be unintuitive to students, but understanding how to use the natural logarithm to undo exponential functions is key in various fields including finance, biology, and physics.

Compound interest is the process of generating earnings on an asset's reinvested earnings. To work out the compound interest, you use the formula mentioned earlier in exponential equations. In essence, compound interest is interest on interest and will make a sum grow at a faster rate than simple interest, which is only calculated on the principal amount.
This concept is crucial for financial growth over time. For example, in the original exercise, the annual percentage yield (APY) of 3.1% indicates how much money you would earn from interest in one year, factoring in compounding. The formula for compound interest, assuming the interest is compounded once per year, can be simplified to:
\[A(t) = A_0 \times (1 + r)^t\]
This shows that the balance of the savings account after a certain number of years is a function of the initial principal (the original deposit), the interest rate, and the number of times that interest is compounded per unit \(t\), which is annually in this case.
Compound interest is the reason why it's important to start saving early, as the effects of compounding can significantly increase the value of savings over time. That is why this concept is often referred to as 'the eighth wonder of the world' by financiers, highlighting the exponential growth potential of investments over time.