\( g ( x ) = \log _ { b } x \quad x = b ^ { - 3 } \)

Did you know that logarithmic functions can trace their roots back to the early 17th century when mathematicians like John Napier started using them to simplify complex calculations? These groundbreaking tools allowed scientists and navigators of the time to perform multiplication and division more easily, making it essential for advancements in fields like astronomy and engineering. When using the given equation \( g(x) = \log_b x \) with \( x = b^{-3} \), you can simplify it to \( g(b^{-3}) = \log_b (b^{-3})\). This becomes \( g(b^{-3}) = -3 \) since logarithms express how many times to multiply the base \( b \) to achieve \( b^{-3} \). Essentially, this means that you are working with a powerful calculator that dives deeper into the relationship between exponents and logarithms!