Solve the equation. \( \log _{9} x=\frac{1}{2} \) A. 0.002 B. 3 C. 81 D. 512

To solve the equation \( \log_{9} x = \frac{1}{2} \), we start by rewriting it in exponential form. This gives us \( x = 9^{\frac{1}{2}} \). Since \( 9 = 3^2 \), we can substitute to get \( x = (3^2)^{\frac{1}{2}} = 3^{2 \cdot \frac{1}{2}} = 3^1 = 3 \). Thus, the answer is B. 3. If you're curious about logarithms, they have a fascinating history! The concept, which dates back to the early 17th century, was developed by John Napier, whose work revolutionized calculations and helped pave the way for modern mathematics. Logarithms transformed multiplication into addition, making complex calculations much simpler and boosting advancements in science and engineering. In practical applications, logarithms are incredibly useful, especially in fields like acoustics (measuring decibels), finance (calculating compound interest), and even in social sciences (analyzing data distributions). For example, the Richter scale for earthquakes is logarithmic, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude, illustrating just how powerful and essential logarithmic calculations are in real life!