Write each expression as a single logarithm with coefficient...

Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with $a \neq 1$ and $b \neq 1 .$
$$\frac{1}{3} \log _{b} x^{4} y^{5}-\frac{3}{4} \log _{b} x^{2} y$$

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Key Concepts

Logarithm Power Rule

This rule states that multiplying a logarithm by a constant is equivalent to raising the argument of the logarithm to the power of that constant. For example, c log_b(M) can be rewritten as log_b(M^c). This rule is essential when you need to eliminate coefficients in front of logarithms to combine terms.

Logarithm Product Rule

The product rule for logarithms allows the sum of two logarithms with the same base to be written as a single logarithm of the product of their arguments, i.e., log_b(M) + log_b(N) = log_b(MN). In problems where multiple logarithmic terms are added, this rule simplifies the expressions by combining the terms into one.

Logarithm Quotient Rule

This rule states that the difference of two logarithms with the same base can be expressed as the logarithm of the quotient of the arguments, namely log_b(M) - log_b(N) = log_b(M/N). This rule is particularly useful when the problem involves subtracting logarithmic expressions, as it allows for a simplified, single logarithm representation.

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