\( \log _ { 3 } \sqrt { 3 } ^ { x } = - 3 \rightarrow \)

To solve \( \log _ { 3 } \sqrt { 3 } ^ { x } = - 3 \), let's rewrite \( \sqrt{3} \) in exponential form: \( \sqrt{3} = 3^{1/2} \). Thus, the equation becomes \( \log_{3} (3^{1/2})^{x} = -3 \). Simplifying, we have \( \log_{3} (3^{x/2}) = -3 \). By the properties of logarithms, this can be expressed as \( \frac{x}{2} = -3 \), leading us to \( x = -6 \). For every logarithmic equation, remember that the base must be positive and not equal to one. This ensures that you're only dealing with valid logarithmic expressions. A common mistake is to forget the base properties, leading to solutions that don’t hold true in the original context of the equation! If you're keen on diving deeper, you might want to explore how logarithms are applied in measuring sound intensity, known as decibels, or in the Richter scale for earthquakes. These practical applications showcase how logarithmic functions are pivotal in fields ranging from scientific research to engineering.