In Exercises 41-70 , use properties of logarithms to...

In Exercises $41-70$ , use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1 .$ Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right] $$

In Exercises $41-70$ , use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1 .$ Where possible, evaluate logarithmic expressions without using a calculator.
$$
\frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right]
$$

Key Concepts

Logarithm Power Rule

The logarithm power rule states that multiplying a logarithm by a constant is equivalent to raising its argument to the power of that constant. In other words, k·ln(a) can be rewritten as ln(a^k). This property is essential when condensing logarithmic expressions, as it allows coefficients in front of logarithms to be incorporated into the argument of the logarithm itself.

Logarithm Product Rule

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors; formally, ln(a) + ln(b) = ln(ab). This rule is crucial for combining separate logarithmic terms into a single log expression when the terms represent multiplicative factors.

Logarithm Quotient Rule

The quotient rule of logarithms expresses that the logarithm of a quotient can be written as the difference of the logarithms of the numerator and denominator: ln(a) - ln(b) = ln(a/b). This property is used to combine logarithms that are subtracted from one another into a single logarithm that represents a division of their arguments.

Algebraic Factorization

Algebraic factorization involves rewriting expressions inside a logarithm in a factored form to simplify the expression further. For example, recognizing common algebraic identities such as the difference of squares allows for further simplification and cancellation when combined with logarithmic properties, leading to a more compact final expression.

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