3. Which is equivalent to the expression below? \( \log \left(16 x^{12}\right) \) \( \begin{array}{ll}\text { A. } 4 \log \left(2 x^{3}\right) & \text { C. } \log \left(2 x^{12}\right)+\log \left(14 x^{12}\right) \\ \text { B. } \log \left(\frac{32 x^{20}}{2 x^{8}}\right) & \text { D. } \log \left(\frac{32 x^{24}}{2 x^{2}}\right)\end{array} \)

Let's break down the expression \( \log \left(16 x^{12}\right) \) using properties of logarithms. We can apply the product property: \( \log(ab) = \log a + \log b \). So, we have: \[ \log(16) + \log(x^{12}). \] Now, \( \log(x^{12}) \) can be simplified using the power property: \( \log(a^b) = b \log a \). Thus, we get: \[ \log(16) + 12 \log(x). \] Next, let's express \( \log(16) \) as \( \log(2^4) = 4 \log(2) \). Putting it all together, we arrive at: \[ 4 \log(2) + 12 \log(x) = 4 \log(2) + 4 \cdot 3 \log(x) = 4(\log(2) + 3 \log(x)). \] Now, evaluating the given options, option A: \( 4 \log(2 x^{3}) \) simplifies to: \[ 4(\log(2) + \log(x^3)) = 4(\log(2) + 3\log(x)), \] which matches our derived expression! So, the equivalent expression is option A! 1. Logarithms have a fascinating history rooted in the work of ancient mathematicians. They were initially developed for simplifying complex calculations before calculators took over. John Napier introduced them in the 16th century, which revolutionized mathematics and navigation, making computations less tedious. It’s pretty amazing to think about how these concepts shaped our modern understanding of numbers! 2. In the real world, logarithms are crucial in fields like science, finance, and engineering. They help model exponential growth, like population trends or compound interest. Engineers use logarithms for sound intensity calculations, while data scientists often rely on them for scaling features in machine learning. Understanding how to manipulate logs opens the door to solving real problems and making sense of complex data in everyday life!