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Understanding logarithmic identities is fundamental when simplifying complex logarithmic expressions. These identities help you manipulate and transform logarithmic expressions into more manageable forms.
One of the main identities includes the log of a quotient, log of a product, and the power rule.
These are helpful especially when dealing with expressions involving roots and exponents, as they allow you to break them down into smaller, simpler pieces.

• The identity for the logarithm of a product states that the log of a product is equal to the sum of the logs: \( \log(AB) = \log(A) + \log(B) \).
• For a quotient, the identity is: \( \log\left(\frac{A}{B}\right) = \log(A) - \log(B) \).
• The power rule asserts that \( \log(A^b) = b \cdot \log(A) \), meaning we can move exponents in front of the log as a multiplier.

These identities are the backbone of simplifying any logarithmic expression, helping us convert complicated expressions into something easier to handle.

The quotient rule for logarithms helps us deal with the logarithm of a division of two numbers. According to this rule, the log of a quotient is the difference between the log of the numerator and the log of the denominator.
In the exercise above, we started by applying this rule to break the expression into two simpler parts. For example, given an expression \( \log\left(\frac{A}{B}\right) \), we express it as:

• \( \log\left(\frac{A}{B}\right) = \log(A) - \log(B) \)

This way, we separate the components of the fraction into smaller parts, which can be simplified individually using other rules like the power and product rules. This step is especially crucial in handling exponents and roots as seen in this problem.

The power rule for logarithms is a powerful tool allowing us to simplify log expressions that involve exponents. It states that the log of a number raised to an exponent is the exponent times the log of the base number.
For instance, if you have \( \log(A^b) \), this can be simplified to:

• \( \log(A^b) = b \cdot \log(A) \)

In the context of the exercise, we applied this rule twice: once to the square root \( \sqrt{y} \), which is expressed as \( y^{1/2} \), becoming \( \frac{1}{2} \log(y) \).
Additionally, it was applied to the cube root \( \sqrt[3]{z} \) expressed as \( z^{1/3} \), resulting in \( \frac{1}{3} \log(z) \).
This rule allows us to handle expressions with roots and powers systematically, making it easier to work through complex logarithmic expressions.

The product rule for logarithms assists with understanding logs of products and breaking them into sums of logs. According to this rule, the log of a product is the sum of the logs of its factors.
In mathematical terms, \( \log(AB) \) becomes

• \( \log(A) + \log(B) \).

In our problem, the product rule helps us manage the expression \( \log(x^4 \sqrt[3]{z}) \) by breaking it into two parts as \( \log(x^4) + \log\sqrt[3]{z} \).
This was pivotal for further simplification with the power rule.
Using the product rule correctly allows for step-by-step simplification of expressions that contain multiple multiplicative terms, streamlining the process and making complex expressions more approachable.