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Exponential equations are equations where unknown variables appear as exponents. They often have the general form \( a^x = b \), where \( a \) and \( b \) are constants, and \( x \) is the variable to be solved. When dealing with exponential equations, especially in contexts like compound interest or population growth, it is common to encounter the natural function form \((e^x)\) or other bases like our given form \((1.02)^t\). To solve for the variable, our main task is to "bring down" the exponent, transforming the equation into a more manageable algebraic form. In many cases, utilizing logarithms allows us to do this effectively. By applying the logarithm to both sides of the equation, we change the multiplicative nature of exponentiation into an additive process that can be calculated straightforwardly.

Logarithms are fundamental tools in mathematics, especially when dealing with exponential equations. They help convert exponential terms into linear terms which are easier to handle.One of the key properties of logarithms is the **power rule**: \( \ln(a^b) = b \cdot \ln(a) \). This property is critical as it allows the exponent to be manipulated as a coefficient in our equations. When you take the natural logarithm of both sides, you're effectively undoing the exponent. For instance, in our solution, taking \( \ln((1.02)^t) \) allows us to bring \( t \) down front, creating an equation we can solve. Alongside the power rule, other useful properties include:- **Product Rule:** \( \ln(xy) = \ln(x) + \ln(y) \)- **Quotient Rule:** \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \)These properties simplify complex equations, conveying exponential relationships into a form where coefficient-based algebra becomes applicable.

Solving equations, especially exponential types, revolves around logical transformations that allow isolating the desired variable. In the case of our example, the main goal was to find \( t \).First, applying the natural logarithm to both sides \( \ln(2) = \ln((1.02)^t) \) made it simpler by converting the exponential to a linear form due to the power rule. This gave us \( \ln(2) = t \cdot \ln(1.02) \). Next, solving for \( t \) means we need to isolate it. Dividing both sides by \( \ln(1.02) \) helps isolate \( t \):- \( t = \frac{\ln(2)}{\ln(1.02)} \).This step involves basic algebra: divide both sides of the equation by the coefficient of \( t \). Using numerical tools like a calculator to approximate logarithmic values (\(\ln(2) \approx 0.6931\) and \(\ln(1.02) \approx 0.0198\)) allows for finding an approximate value for \( t \), which in this solved exercise turns out to be approximately \(35.0151\). The logical approach of isolating and solving these transformed equations remains consistent across various problems, marking a reliable method in dealing with such mathematical challenges.