91Ó°ÊÓ

Logarithms have unique properties that simplify calculations. One useful property is the power rule, expressed as: \[a \times \text{ln}\thinspace b = \text{ln}(b^a) \]
This rule lets us combine multiplication and logarithms into one logarithmic expression. For instance, consider \(\frac{1}{2} \text{ln}\thinspace 16 \). Using the power rule, we transform it into \(\text{ln}(16^{1/2})\). This simplifies further to \(\text{ln}(\text{sqrt}(16))\), which is \(\text{ln}\thinspace 4\).
Similarly, for \(\frac{1}{3} \text{ln}\thinspace 27 \), applying the power rule converts it into \(\text{ln}(27^{1/3})\). This becomes \(\text{ln}(\text{sqrt[3]}{27})\), simply \(\text{ln}\thinspace 3\).
These properties help make complex logarithmic problems much easier to handle.

Comparing logarithmic values is straightforward if you know their properties. After simplifying, you end up with \(\text{ln}\thinspace 4\) and \(\text{ln}\thinspace 3\). Logarithms have a base (commonly 'e' or 10) which makes such comparisons valid if the bases are the same.
Because 4 is greater than 3, it follows that \(\text{ln}\thinspace 4 \) must be greater than \(\text{ln}\thinspace 3\). This holds because logarithms are monotonic functions, meaning they preserve the order.
So, to compare \(\frac{1}{2} \text{ln}\thinspace 16\) and \(\frac{1}{3} \text{ln}\thinspace 27\), you only need to compare \(\text{ln}\thinspace 4\) and \(\text{ln}\thinspace 3\).

A function is called monotonic if it is either entirely non-increasing or non-decreasing. The natural logarithm function, \( \text{ln}(x) \), is a strictly increasing function. This means if \( x_1 < x_2 \), then \( \text{ln}(x_1) < \text{ln}(x_2) \).
This property simplifies the comparison of logarithmic expressions. For \( \text{ln}\thinspace 4 \) and \( \text{ln}\thinspace 3 \), since 4 is greater than 3, \( \text{ln}\thinspace 4 \) is greater than \( \text{ln}\thinspace 3 \).
In summary, the monotonic nature of \( \text{ln}(x) \) ensures that the order of numbers is preserved, so larger inputs give larger logarithmic values.