\( \begin{array}{l}\text { Solve for } x \\ \begin{array}{ll}6.1\end{array} \quad 5^{x}=\frac{1}{5} \\ 6.2 \\ 6.3 .\end{array} \frac{x+1)=x+4}{2}=\frac{2 x+7}{3} \quad \) (5)

To solve \(5^x = \frac{1}{5}\), we can rewrite \(\frac{1}{5}\) as \(5^{-1}\). This gives us \(5^x = 5^{-1}\). Since the bases are the same, we can set the exponents equal to each other: \(x = -1\). For the equation \(\frac{x+1}{2} = \frac{2x+7}{3}\), start by cross-multiplying to eliminate the fractions: \(3(x + 1) = 2(2x + 7)\). Expand both sides: \(3x + 3 = 4x + 14\). Rearranging gives: \(3 = x + 14\) or \(x = -11\). Now you have \(x = -1\) and \(x = -11\)! You know what’s fun? Exponentials and fractions have a long and wild history! The concept of logarithms, which are the opposite of exponentials, was developed in the 16th century by John Napier. He basically made life easier for scientists and mathematicians by turning complex multiplication and division into simpler addition and subtraction. A true game changer! Real-world application? You bet! Exponentials pop up everywhere, from finance to biology. For example, if you invest money with compound interest, you’re diving into the world of exponential growth. Similarly, populations of organisms can grow exponentially under ideal conditions. Just think of how quickly bacteria can multiply—one cell could turn into millions in no time!