Solve the exponential equation. Express irrational solutions in exact form. \( 5\left(9^{6 \mathrm{x}}\right)=9 \) What is the exact answer? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \). (Simplify your answer. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression. Type an exad B. There is no solution.

To solve the equation \( 5\left(9^{6x}\right) = 9 \), we first divide both sides by 5, yielding \( 9^{6x} = \frac{9}{5} \). Next, we can express 9 as \( 3^2 \), which gives us \( (3^2)^{6x} = \frac{3^2}{5} \). This simplifies to \( 3^{12x} = \frac{3^2}{5} \). By converting to logarithm form or matching bases, we end up with \( 12x = 2 - \log_3(5) \). Solving for \( x \), we have \( x = \frac{1}{6} - \frac{\log_3(5)}{12} \). The exact solution thus is \( \left\{ \frac{1}{6} - \frac{\log_3(5)}{12} \right\} \). The fascinating part of logarithms is how they grew from the need to simplify complex calculations, especially in navigation and science, during the 16th century. They transformed the way mathematicians and astronomers approached problems, leading to more accurate calculations and explorations! Many real-world applications make use of exponential equations, from calculating compound interest in finance to population growth in ecology. Understanding these principles helps us predict future scenarios, making decisions based on mathematical models in various fields—like deciding how much money to invest over time for a fruitful retirement, or analyzing the spread of a virus. Keeping these equations and their implications in mind can be truly enlightening!