Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logbx²

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Recognize that the expression \( \log_b x^2 \) can be simplified using the power rule of logarithms, which states \( \log_b (x^n) = n \cdot \log_b x \).

Apply the power rule to the expression: \( \log_b x^2 = 2 \cdot \log_b x \).

Substitute the given value of \( \log_b x = 0.36 \) into the expression: \( 2 \cdot 0.36 \).

Perform the multiplication to simplify the expression: \( 2 \cdot 0.36 \).

The result of the multiplication gives the value of \( \log_b x^2 \).

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주요 개념

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Properties of Logarithms

Logarithms have several key properties that simplify calculations. The most relevant ones include the product property (log_b(mn) = log_b(m) + log_b(n)), the quotient property (log_b(m/n) = log_b(m) - log_b(n)), and the power property (log_b(m^n) = n * log_b(m)). Understanding these properties is essential for manipulating logarithmic expressions effectively.

Power Property of Logarithms

The power property states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of the base number. For example, log_b(x^n) = n * log_b(x). This property is particularly useful when evaluating expressions involving powers, as it allows for simplification before calculation.

Base of Logarithm

The base of a logarithm indicates the number that is raised to a power to obtain a given value. In the expression log_b(x), 'b' is the base. Understanding the base is crucial because it affects the value of the logarithm and the application of logarithmic properties. In this question, knowing the base helps in evaluating log_b(x^2) using the power property.

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.logbx²