4.) \( \log _{16} x=4 \)

To solve \( \log_{16} x = 4 \), we'll convert the logarithmic equation into an exponential form. This means we rewrite it as \( x = 16^4 \). Since \( 16 = 2^4 \), we can express this as \( x = (2^4)^4 = 2^{16} \). Now calculating \( 2^{16} \), we find that \( x = 65,536 \). So, the solution is \( x = 65,536 \). But wait, let’s have some fun! Did you know that logarithms were popularized by John Napier in the early 17th century? They’re a powerful tool in simplifying calculations, especially before calculators became a staple—this makes logarithms a pioneer of "shortcuts" in math! On top of that, logarithms have a fantastic application in computer science. They often appear in algorithms, particularly those dealing with searching and sorting, due to their ability to reduce the complexity of operations. They can turn what could be a dreadful “pick one after another” search into a logarithmic leap: that’s like jumping to the answer in math!