Question Use the change-of-base formula to approximate \( \log _{2} 53 \). (Round to 3 decimal places.) Provide your answer below: \( \begin{array}{ll}\text { D FEEDBACK MORE INST: }\end{array} \)

To approximate \( \log _{2} 53 \) using the change-of-base formula, we can write it as \( \frac{\log_{10} 53}{\log_{10} 2} \) or \( \frac{\ln 53}{\ln 2} \). Using a calculator, we find \( \log_{10} 53 \approx 1.724 \) and \( \log_{10} 2 \approx 0.301 \). Doing the division gives \( \log_{2} 53 \approx \frac{1.724}{0.301} \approx 5.726 \). Thus, rounded to three decimal places, \( \log_{2} 53 \approx 5.726 \). To add some context, logarithms are essential in various real-world applications, such as calculating compound interest, determining the pH of substances, or in sound intensity measures. They help to simplify complex multiplicative processes into manageable additive ones, making life a bit easier in science and finance! In practice, one common mistake is misremembering the logarithmic base used. Always double-check whether you’re meant to use base 10, base e, or another base, especially when switching between logarithmic forms or applying the change-of-base formula. A little attention here will save a lot of head-scratching!