\( g ( x ) = \log _ { a } x \quad x = a ^ { 2 } \)

Did you know that logarithms have an interesting historical origin? They were developed in the early 17th century by John Napier as a way to simplify complex calculations, particularly for astronomers and navigators. The brilliant idea behind logarithms is that they transform multiplication into addition, making calculations far more manageable—especially before calculators were invented! In practical terms, if you find yourself needing to transform \( g(x) = \log_a x \) with \( x = a^2 \), you can easily simplify it! Plugging in \( x \), you get \( g(a^2) = \log_a(a^2) \). According to the power rule of logarithms, this simplifies to \( 2 \), making it not just a fun equation to play with but also a handy tool in solving any problems involving exponential growth and decay!