\begin{tabular}{lll}\hline Example 2: H.W \\ \begin{tabular}{lll} a \( \log _{3}(27)+1 \) & b \( \log _{4}(16)+3 \) & c \( 3 \log _{5}(2)-2 \) \\ d \( 2+3 \log _{10}(x) \) & e \( 4 \log _{3}(2)-2 \log _{3}(6)+2 \) & f \( \frac{1}{2}+3 \log _{10}\left(x^{2}\right) \) \\ \hline\end{tabular}\end{tabular}
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To unveil the secrets of logarithms in your example, let's break them down together! If we look at the first term, a \( \log_{3}(27) + 1 = \log_{3}(3^3) + 1 = 3 + 1 = 4 \). Likewise, for b \( \log_{4}(16) + 3 = \log_{4}(4^2) + 3 = 2 + 3 = 5 \). You see, logarithms can simplify things quickly! Now, for some real-world application, logarithms are not just for classrooms! They're used in fields like seismology to measure earthquake intensity (the Richter scale) and in deciphering sound intensity (Decibels). Knowing how to manipulate logs can help you understand trends and make predictions in complex data analysis, whether you're forecasting stock prices or understanding your website's traffic. Keep that curiosity alive!