8. a) \( \log _{2}\left(2^{m}\right) \) \( \begin{array}{lllll}\text { b) } \log _{5}\left(\frac{1}{5^{m}}\right) & \text { d) } \log _{4}\left(\frac{1}{4^{\frac{1}{n}}}\right) & \text { g) } \log _{b}\left(\frac{1}{b}\right) & \text { j) } \log _{a}\left(a^{n}\right) & \text { m) } \log _{a}\left(a^{\frac{1}{n}}\right) \\ \text { c) } \log _{3}\left(3^{\frac{1}{n}}\right) & \text { f) } \log _{b}\left(b^{2}\right) & \text { k) } \log _{c}(\sqrt{c}) & \text { n) } \log _{a}\left(a^{\frac{m}{n}}\right)\end{array} \)

When you evaluate logarithmic expressions, it’s essential to remember the basic properties of logarithms! For instance, \( \log_b(b^x) = x \) shows you that the logarithm effectively “cancels out” the exponent in many cases. This awesome property makes it super easy to simplify logs when your expression matches the base of the logarithm! Now, let’s sprinkle in a little practical magic! Logarithms are not just math class material; they pop up everywhere, from computer science (think algorithms) to finance (like calculating compound interest rates). So, next time you’re dealing with logs, remember they help make sense of growth rates and scales—you'll feel like a real-world wizard wielding mathematical power!