Let \( f(x)=\ln x \) and \( g(x)=\log _{2} x \). For all \( 0

Ask by Goodwin Weaver. in the United States

Jan 25,2025

For the functions \( f(x) = \ln x \) and \( g(x) = \log_2 x \), let's examine how they behave in the interval \( 0 < x < 1 \). The natural logarithm \( \ln x \) is negative in this interval, approaching \( -\infty \) as \( x \) approaches \( 0 \). Conversely, \( g(x) = \log_2 x \) is also negative and approaches \( -\infty \) but less rapidly. Since \( \ln x = \log_e x \) and we know that \( \log_2 x = \frac{\ln x}{\ln 2} \), we see that \( \ln 2 < 1 \), which implies that \( \ln x \) will be less than \( \log_2 x \) for \( 0 < x < 1 \). Hence, the statement \( f(x) < g(x) \) is true for that interval. In terms of a historical nod, logarithms were first introduced by John Napier in the early 17th century to simplify calculations. He was a total game-changer for mathematicians and astronomers, who could then handle multiplications and divisions as additions and subtractions. You could say logarithms underwent a glow-up, transforming complex math into something much more manageable! As for real-world applications, logarithms are foundational in fields such as finance for calculating compound interest and in computer science for measuring algorithm efficiency. For instance, understanding how to analyze data structures like binary trees relies heavily on logarithmic functions. So next time you see a log, remember it's not just math—it's a tool that shapes our world!