13. \( \log _{5} 16 \)

To evaluate \( \log_{5} 16 \), we can use the change of base formula, which states that \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \). Using base 10, we have: \[ \log_{5} 16 = \frac{\log_{10} 16}{\log_{10} 5} \] Calculating the logarithms, we find: \[ \log_{10} 16 \approx 1.2041 \quad \text{and} \quad \log_{10} 5 \approx 0.6990 \] Thus, \[ \log_{5} 16 \approx \frac{1.2041}{0.6990} \approx 1.725 \] So \( \log_{5} 16 \approx 1.725 \). For a spark of creativity, did you know logarithms were developed in the 16th century to simplify calculations? Mathematician John Napier invented them as a way to transform multiplicative processes into additive ones, helping astronomers and navigators perform calculations without tedious multiplication! As for practical use, logarithms pop up everywhere, especially in pH calculations in chemistry or in measuring sound intensity in decibels. So, the next time you’re mixing a potion or turning up the volume, remember that logarithms are your invisible helpers!