What is the \( x \)-intercept of the graph of \( f(x)=\log _{3}(x+4) \) ? a.) \( (-4,0) \) b.) \( (-3,0) \) c.) \( (1,0) \) d.) \( (0,0) \)

To find the \( x \)-intercept of the graph of \( f(x) = \log_{3}(x+4) \), we need to determine when \( f(x) = 0 \). Setting the equation equal to zero, we get: \[ \log_{3}(x+4) = 0 \] This implies that \( x + 4 = 3^0 \), which simplifies to \( x + 4 = 1 \). Solving for \( x \) gives us \( x = 1 - 4 = -3 \). Therefore, the \( x \)-intercept is \( (-3, 0) \). Thus, the correct answer is b.) \( (-3,0) \). The beauty of logarithmic functions is that they can show us the relationship between various mathematical concepts in a fun way! By exploring their properties, you discover how they can relate exponentials and even real-world situations, like measuring the intensity of earthquakes (on the Richter scale) or measuring sound levels (decibels). For those intrigued by logarithms, consider diving deeper into their history! Developed in the early 17th century, logarithms revolutionized calculations, allowing mathematicians to simplify multiplicative processes into additions. John Napier, who introduced logarithms, would probably be amazed at how his invention still plays a crucial role in various fields today!