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Logarithms are a powerful mathematical tool that allow us to work with exponents in a more manageable way. One important logarithmic rule to understand is the identity rule: \( \log_{b} 1 = 0 \). This means that the logarithm of 1 to any base \( b eq 1 \) is always 0.
This is because any number raised to the power of 0 equals 1. Hence, when you encounter a problem like \( \log_{1.5} 1 \), the answer is straightforwardly 0.
This identity rule is a fundamental concept that students should remember as it simplifies many logarithmic expressions.

• \( \log_{10} 1 = 0 \)
• \( \log_{2} 1 = 0 \)
• \( \log_{b} 1 = 0 \)

Understanding these rules not only helps compute logarithms faster but also builds a strong foundation for more complex logarithmic calculations.

The base of a logarithm is a crucial component that students must grasp. In simple terms, the base \( b \) in a logarithm \( \log_{b} x \) refers to the number that we repeatedly multiply by itself. For example, in the expression \( \log_{2} 8 \), the base is 2. This indicates that 2 is raised to a certain power to get 8.
In our exercise, the base is 1.5, which means we are looking for the power to which 1.5 must be raised to result in 1.
Selecting the proper base is essential because it determines the whole structure of the logarithmic function.

• The base of a logarithm is always positive and not equal to 1.
• It shows us the root or the exponent step in exponential functions.

This clear understanding of base allows students to manipulate logarithmic functions more easily and recognize the intricacies of various logarithmic behaviors.

Evaluating logarithms involves finding the power or exponent that the base of the logarithm must be raised to yield a given number.
To evaluate a logarithm like \( \log_{b} x \), you need to solve the equation \( b^{y} = x \), where \( y \) is what you're solving for.
Let's take the example from our exercise: evaluate \( \log_{1.5} 1 \). Here, you're essentially trying to find what power 1.5 must be raised to get 1. As per the rule, \( \log_{b} 1 \) equals 0, hence, \( 1.5^{0} = 1 \).
Here are some steps to consider when evaluating logarithms:

• Recognize the base and the number you're evaluating.
• Check to see if any logarithm rules can simplify the process.

By mastering these steps, you can easily evaluate logarithms, making it easier to tackle more complex mathematical problems down the line.