Use the properties of logarithms to write each expression as...

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1.
$$
\log _{a} m-\log _{a} n
$$

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Key Concepts

Quotient Rule of Logarithms

The quotient rule for logarithms states that the difference between two logarithms with the same base is equal to the logarithm of the quotient of their arguments. In mathematical terms, log_b(x) - log_b(y) is equivalent to log_b(x/y), which is a key identity used to condense logarithmic expressions into a single logarithm.

Domain Considerations for Logarithms

Logarithmic functions are defined only for positive arguments, and the base of a logarithm must be positive and not equal to one. These domain restrictions are essential to ensure that logarithmic expressions are valid and meaningful when applying the properties of logarithms.

Properties of Logarithms

Logarithms have several useful properties, such as the product rule, quotient rule, and power rule, which allow for the simplification and manipulation of logarithmic expressions. These properties are fundamental in algebra and are used to combine or break apart logarithms logically based on their operations.

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