91Ó°ÊÓ

The product rule is a fundamental property of logarithms. It states that the logarithm of a product is the sum of the logarithms of the factors. Formally, for any positive numbers \(a\) and \(b\) and any logarithm base \(c\), the product rule is written as \[\log_{c} (a) + \log_{c} (b) = \log_{c} (a \cdot b)\].
This rule is extremely useful when you need to combine multiple logarithmic terms into one. For example, if you have \(\log_{b} 65\) and \(\log_{b} 2\), you can use the product rule to combine them: \[\log_{b} 65 + \log_{b} 2 = \log_{b} (65 \cdot 2)\].
By applying this rule correctly, you can simplify many logarithmic expressions easily. The key is to remember that the logarithms must share the same base for the product rule to apply. If the bases are different, this rule cannot be used directly.

Transforming multiple logarithmic expressions into a single logarithm is often necessary for simplification. To achieve this, we rely on the logarithmic properties such as the product rule.
In the given exercise, we started with two logarithms: \(\log_{b} 65\) and \(\log_{b} 2\). Using the product rule, we combined them into one: \[\log_{b} (65 \cdot 2)\].
Next, simplifying the expression inside the logarithm by performing the multiplication: \[65 \cdot 2 = 130\].
Thus, the expression as a single logarithm becomes: \[\log_{b} 130\].
Breaking down expressions into single logarithms makes them easier to work with, especially in complex equations or when solving for unknowns.

Logarithmic expressions are equations involving logarithms. These expressions can sometimes look complex, but they can often be simplified using logarithm properties like the product rule, quotient rule, and power rule.
Consider the expression from the exercise: \[\log_{b} 65 + \log_{b} 2\].
You can recognize that both terms share the same base \(b\). By applying the product rule, you combine them into a single expression: \[\log_{b} 130\].
Simplifying logarithmic expressions often involves combining terms or breaking them apart using the appropriate rules. Recognizing the properties of logarithms helps in solving complex problems and can make further calculations more manageable.
Remember, always look for ways to apply these properties to make your logarithmic expressions simpler and easier to work with.