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. $$ \frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right] $$

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.
$$
\frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right]
$$

Key Concepts

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves using the aforementioned logarithmic properties to combine several logarithm terms into one. This process simplifies the overall expression, reduces complexity, and makes further manipulation or evaluation more straightforward.

Quotient Rule

The quotient rule explains that the difference of two logarithms with the same base can be expressed as the logarithm of a quotient, given by ln(a) - ln(b) = ln(a/b). This rule is key to combining logarithmic expressions where subtraction is involved.

Power Rule

The power rule states that multiplying a logarithm by a constant is equivalent to raising the argument of the logarithm to that constant power, expressed as b·ln(a) = ln(a^b). This rule is essential when dealing with coefficients in front of logarithms, enabling the rewriting of such expressions in a form that is easier to combine or simplify.

Logarithm Properties

Logarithm properties are a set of fundamental rules that allow the manipulation and simplification of logarithmic expressions. These rules, including the power, product, and quotient rules, enable one to combine multiple logarithms into a single logarithm or expand a single logarithm into multiple terms, facilitating both simplification and evaluation.

Product Rule

The product rule indicates that the sum of two logarithms with the same base can be combined into a single logarithm of the product of their arguments, written as ln(a) + ln(b) = ln(ab). This rule is especially useful when condensing multiple logarithmic terms into one.

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