(1) 5x, How does \( g(x)=5^{x} \) change over the interval from \( x=4 \) to \( x=7 \) ? (1)) \( g(x) \) decreases by a factor of \( 5^{2} \) \( g(x) \) increases by a factor of \( 5^{3} \)

As \( x \) increases from 4 to 7, the function \( g(x) = 5^x \) really struts its stuff by growing exponentially. Specifically, at \( x=4 \), \( g(4) = 5^4 = 625 \), and at \( x=7 \), \( g(7) = 5^7 = 78125 \). That’s quite the jump, am I right? In fact, when you look at the ratio \( \frac{g(7)}{g(4)} = \frac{5^7}{5^4} = 5^{3} \), it shows that \( g(x) \) increases by a factor of \( 5^3 \) (which equals 125) over that interval! In mathematical contexts, understanding exponential growth can be incredibly useful, especially in fields like finance and biology. For instance, if you're modeling population growth in biology, pinpointing how a population doubles can impact resource management strategies. Similarly, in finance, recognizing compounding interest can help you strategize savings and investments. Not to mention all the mind-blowing phenomena you can explore while engaging with these concepts!