Write the expressions as a single logarithm and simplify if...

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Key Concepts

Logarithm Power Rule

This rule states that a constant multiplied by a logarithm can be rewritten as an exponent on the argument of the logarithm. For example, c · log_b(x) can be expressed as log_b(x^c). This is particularly useful when coefficients such as fractions or negatives need to be incorporated into the argument for further simplification.

Logarithm Quotient Rule

When subtracting one logarithm from another with the same base, the expressions can be combined into a single logarithm representing the division of the arguments. That is, log_b(M) - log_b(N) equals log_b(M/N). This rule is crucial for merging different logarithmic terms into one.

Logarithm Product Rule

This property allows the addition of two logarithms with the same base to be written as a single logarithm of the product of their arguments. Specifically, log_b(M) + log_b(N) equals log_b(M·N). It is often used alongside the power and quotient rules to consolidate multiple logarithmic expressions.

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