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.
$$
4 \ln (x+6)-3 \ln x
$$

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Key Concepts

Properties of Logarithms

Properties of logarithms are fundamental rules that enable the manipulation and simplification of logarithmic expressions. They include the product, quotient, and power rules, and are essential for tasks such as condensing multiple logarithm terms into a single logarithm.

Power Rule

The power rule for logarithms allows you to rewrite a logarithm with a coefficient by moving the coefficient as an exponent inside the logarithm. This transformation is crucial when condensing expressions because it converts multiplication outside the log to an exponentiation within it.

Quotient Rule

The quotient rule states that the difference between two logarithms of the same base is equivalent to the logarithm of the quotient of their arguments. This property is used to combine multiple logarithmic terms into a single logarithm by turning a subtraction into a division within the logarithm.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves using the logarithm properties — notably the power and quotient rules — to combine several logarithmic terms into one. This process simplifies the expression, making it easier to evaluate or further manipulate in algebraic and calculus contexts.

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