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The power rule of logarithms is a significant tool when dealing with logarithmic expressions involving exponents. This rule says that when you have a logarithm of a number raised to a power, you can simplify it by bringing the exponent down in front of the logarithm.

For example, if you have \[\log_b (a^m)\], you can simplify it to \[m \cdot \log_b (a)\]. This can make your calculations much easier.

• The base of the logarithm, represented as \(b\), remains unchanged.
• Only the exponent \(m\), is moved to the front of the logarithmic expression.

In the given exercise, this concept is applied to simplify \[\log_2 (y^{100})\] into \[100 \cdot \log_2 (y).\] This allows us to easily substitute and evaluate the expression. Remembering this property can help tremendously in solving more complex logarithmic problems.

Logarithmic expressions are mathematical phrases that involve logarithms, which are the inverse operations of exponentiation. If you understand exponentiation, understanding logarithms becomes a lot easier.

Logarithms answer the question: "To what power must the base be raised to get a certain number?" For instance, \[\log_b (a) = c\] answers the question: "What power must \(b\) be raised to, to yield \(a\)?"

• Common bases include 10 (common logarithm) and \(e\) (natural logarithm), but in this exercise, the base is 2.
• Expressing logarithms with different bases is possible through the change of base formula, if needed.

Understanding how to work with logarithmic expressions, like substituting values or simplifying powers, is essential in algebra and calculus. This exercise reinforces how substituting known values into these expressions simplifies the evaluation process.

Exponentiation within the context of logarithms often implies the need to apply logarithmic rules like the power rule. In the exercise, you deal with \(y^{100}\), so the target is to simplify using logarithm properties.

Exponentiation within logarithms requires that you carefully manage and simplify expressions. You do this by applying the power rule, which handles the exponentiation in logarithms directly. This allows a more straightforward calculation of \(\log_2 (y^{100})\) and bridges understanding on how exponents interact with logarithms.

• Exponentiating implies multiplication - every time, you multiply the base by itself the number of times indicated by the exponent.
• In logarithmic terms, this multiplication becomes a simple multiplication with the exponent by moving it to the front.

This is an elegant way to deal with potentially cumbersome numbers when calculating values involving both powers and logarithms. It highlights the symbiotic relationship between logs and exponents.