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The power rule of logarithms is a key concept that helps simplify expressions involving logarithms. The rule states that multiplying a logarithm by a constant can be expressed as raising the argument of the logarithm to the power of that constant. Mathematically, this is written as:

• For any real number \(a\) and positive number \(b\), the expression \(a \ln b\) simplifies to \(\ln b^a\).

By applying the power rule, you can transform terms such as \(2 \ln (x+1)\) into \(\ln((x+1)^2)\) and \(\frac{1}{3} \ln x\) into \(\ln(x^{1/3})\).
This simplification makes it easier to handle logarithmic expressions as you work through algebraic manipulations. Remember, this rule is only applicable when dealing with the transformation of coefficients of logarithms into exponents.

The product rule of logarithms is another fundamental tool that facilitates the process of combining multiple logarithmic expressions into one. According to this rule, the sum of two logarithms with the same base can be merged into a single logarithm. Specifically, the expression \(\ln a + \ln b\) becomes \(\ln (a \cdot b)\).
This property is particularly useful in simplifying expressions and finding a single logarithm from multiple terms.
In our given expression, after applying the power rule, you can use the product rule to combine \(\ln((x+1)^2)\) and \(\ln(x^{1/3})\). This results in a more concise expression: \(\ln((x+1)^2 \cdot x^{1/3})\). Keep in mind that this rule helps in multiplying the arguments within the logarithms, allowing you to express compound logarithmic expressions more clearly.

The quotient rule of logarithms allows you to express the difference of two logarithms as the logarithm of a quotient. It is written as \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).
This rule is very handy for simplifying expressions where you have subtraction of logs, turning them into a single log expression involving a division.
In our current problem, after applying the product rule, you end up with an expression like \(\ln((x+1)^2 \cdot x^{1/3})\). By using the quotient rule, you combine it with the term \(-\ln(\cos x)\). The final result is \(\ln \left( \frac{(x+1)^2 \cdot x^{1/3}}{\cos x} \right)\). This simplification is very powerful in algebra, making complex logarithmic expressions more manageable.