Write as a single logarithm. Assume the variables are...

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to $1 .$
$$\frac{1}{2} \log _{5} a-4 \log _{5} b$$

Key Concepts

Quotient Rule of Logarithms

The quotient rule for logarithms indicates that the difference between two logarithms with the same base can be expressed as the logarithm of a quotient. Specifically, log_b(x) - log_b(y) equals log_b(x/y). This rule is useful for combining logarithmic terms into a single logarithm, which simplifies solving problems involving multiple logarithmic expressions.

Domain Restrictions for Logarithms

Since a logarithm is only defined for positive arguments, it is important to ensure that the values inside the logarithm remain positive. In problems involving logarithmic expressions, conditions must be established so that both the base of the logarithm (which must be positive and not equal to 1) and the argument are valid, ensuring the expression is defined.

Logarithmic Functions

Logarithms are the inverse operations of exponentiation, where the logarithm log_b(x) answers the question, 'to what power must the base b be raised to obtain x?' This concept is foundational in many areas of mathematics and is particularly important in solving equations involving exponential functions.

Power Rule of Logarithms

The power rule states that a constant multiplied by a logarithm can be rewritten as the logarithm of the argument raised to that constant’s power; mathematically, c·log_b(x) is equal to log_b(x^c). This rule is essential when condensing expressions involving logarithms that have numerical coefficients.

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