91Ó°ÊÓ

The power rule for logarithms is a fundamental concept that helps simplify expressions involving exponents within a logarithmic function. This rule states that if you have a logarithm of a term raised to a power, you can multiply the power by the logarithm of the base. Mathematically, it is expressed as: - \( \log_b(a^n) = n \log_b(a) \).
This comes in handy when simplifying complex logarithmic expressions, as seen in our exercise. For instance, in the expression \( \log x^{2} \), applying the power rule allows us to write this as \( 2 \log x \).
Similarly, if you're dealing with roots, you can express them as fractional powers. For example, \( \sqrt{y} \) becomes \( y^{1/2} \). Applying the power rule here, \( \log(\sqrt{y}) \) transforms to \( \frac{1}{2} \log y \).
These transformations make it much easier to handle and combine logarithmic expressions.

Another key rule for manipulating logarithmic expressions is the product rule. The product rule simplifies the logarithm of a product into a sum of logarithms, which is often easier to work with. This rule can be stated as: - \( \log_b(abc) = \log_b(a) + \log_b(b) + \log_b(c) \).
In our exercise, applying the product rule to \( \log(x^{2} \sqrt{y} z) \) means breaking it down into smaller, more manageable parts. Specifically, it becomes \( \log(x^{2}) + \log(\sqrt{y}) + \log(z) \).
This step is crucial because it allows us to individually apply other logarithmic rules, like the power rule, to each segment of the expression. By doing so, we ultimately simplify the original expression into a sum of separate logarithmic terms like \( 2 \log x + \frac{1}{2} \log y + \log z \).
Understanding and using the product rule can greatly simplify the process of evaluating or combining multiple logarithmic expressions.

The change of base formula for logarithms is an extremely useful tool in mathematics, especially when you need to evaluate logarithms that are not in a familiar or convenient base. This formula allows us to rewrite a logarithm in terms of two other bases of our choosing. It is expressed as:- \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \).
This is particularly useful for calculators and computational software that primarily handle logarithms in base 10 (common logarithms) or base \( e \) (natural logarithms).
By applying the change of base formula, you are able to express any logarithm in terms of logarithms with these more familiar bases. While our exercise doesn't directly require the change of base formula, understanding it allows you to flexibly work with any logarithm, regardless of its original base.
Being proficient with this formula ensures that you can adapt to any context where different logarithmic bases might be presented, enhancing your problem-solving capabilities in a wide array of mathematical situations.