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Exponential growth refers to an increase in number or size, at a consistently rapid rate when compared to a base value. In the context of compound interest, this growth signifies how investments or loans can grow swiftly over time.

For instance, when you deposit money into a savings account, the amount doesn't just increase linearly by the same amount each year. Instead, the interest earned in one year is added to the principal, and in the following year, interest is earned on the new total, essentially earning 'interest on interest'. This compounding effect causes the account balance to grow exponentially.

The formula that captures this exponential growth due to compound interest is \( A = P \times \big(1 + \frac{r}{n}\big)^{n \times t} \), where \(A\) represents the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (initial sum of money), \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested or borrowed for, in years.

Logarithms are a critical mathematical concept, especially when dealing with exponential growth in financial mathematics. A logarithm answers the question: to what exponent must we raise a specific base number to produce a certain value?

In the scenario of our compound interest problem, logarithms are used to isolate the variable \(t\), which represents time. Because compound interest leads to an exponential equation in terms of time, we need to use logarithms to solve for \(t\). The formula used in the step by step solution is a rearranged version of the compound interest formula, solved for \(t\) using logarithms: \[ t = \frac{ \ln{(\frac{A}{P})} }{n \ln{(1 + \frac{r}{n})}} \].

This equation is derived using the natural logarithm (\ln) – which uses the constant \(e\) as the base – because it simplifies calculations involving continuous growth, and allows us to solve for time directly.

Financial mathematics involves using mathematical techniques to solve financial problems. It includes concepts like valuation, investing, budgeting, and saving. Compound interest is a core concept within this area, and understanding it enables individuals to make more informed financial decisions.

The step by step solution provided for determining when the balance exceeds certain amounts at a given interest rate is an application of financial mathematics. The careful organization of the given information – principal, interest rate, and the number of times the interest is compounded – and the application of algebraic methods and logarithms, highlights the process of analyzing financial situations mathematically.

Practical knowledge of how to apply these concepts ensures that individuals are equipped to plan for goals such as retirement, saving for large purchases, or understanding the implications of taking on a loan. Moreover, this adeptness in financial mathematics can also lead to a greater understanding of more complex financial instruments and markets.