The given expression is:

$\log_b \left( \sqrt[3]{\frac{x^2}{yz}} \right)$

To simplify this logarithmic expression, we follow these steps:

Step 1: Rewrite the cube root using exponents

The cube root can be written as a fractional exponent:

$\sqrt[3]{\frac{x^2}{yz}} = \left( \frac{x^2}{yz} \right)^{\frac{1}{3}}$

Substitute this into the logarithm:

$\log_b \left( \sqrt[3]{\frac{x^2}{yz}} \right) = \log_b \left( \left( \frac{x^2}{yz} \right)^{\frac{1}{3}} \right)$

Step 2: Apply the logarithmic power rule

The logarithmic power rule states that:

$\log_b (a^c) = c \cdot \log_b (a)$

Here, $a = \frac{x^2}{yz}$ and $c = \frac{1}{3}$. Applying the rule:

$\log_b \left( \left( \frac{x^2}{yz} \right)^{\frac{1}{3}} \right) = \frac{1}{3} \log_b \left( \frac{x^2}{yz} \right)$

Step 3: Apply the logarithmic quotient rule

The quotient rule for logarithms states that:

$\log_b \left( \frac{M}{N} \right) = \log_b (M) - \log_b (N)$

Here, $M = x^2$ and $N = yz$. Applying the rule:

$\log_b \left( \frac{x^2}{yz} \right) = \log_b (x^2) - \log_b (yz)$

Step 4: Expand further using the product rule

The product rule for logarithms states that:

$\log_b (MN) = \log_b (M) + \log_b (N)$

Thus, for $\log_b (yz)$:

$\log_b (yz) = \log_b (y) + \log_b (z)$

Substitute back:

$\log_b \left( \frac{x^2}{yz} \right) = \log_b (x^2) - \left[ \log_b (y) + \log_b (z) \right]$

Step 5: Simplify $\log_b (x^2)$

The power rule states $\log_b (x^c) = c \cdot \log_b (x)$. Therefore:

$\log_b (x^2) = 2 \log_b (x)$

Substitute back:

$\log_b \left( \frac{x^2}{yz} \right) = 2 \log_b (x) - \log_b (y) - \log_b (z)$

Step 6: Final simplification

Now, multiply the entire expression by $\frac{1}{3}$ (from Step 2):

[ \log_b \left( \sqrt[3]{\frac{x^2}{yz}} \right) = \frac{1}{3} \left[ 2 \log_b (x) - \log_b (y) - \log_b (z) \right] ]

Distribute $\frac{1}{3}$:

$\log_b \left( \sqrt[3]{\frac{x^2}{yz}} \right) = \frac{2}{3} \log_b (x) - \frac{1}{3} \log_b (y) - \frac{1}{3} \log_b (z)$

Final Answer:

$\log_b \left( \sqrt[3]{\frac{x^2}{yz}} \right) = \frac{2}{3} \log_b (x) - \frac{1}{3} \log_b (y) - \frac{1}{3} \log_b (z)$

Would you like me to explain any part further or provide examples?

Related Questions:

1. How do you apply logarithmic rules to simplify powers and roots?
2. What is the logarithmic quotient rule and how is it used?
3. How do you expand logarithmic expressions involving products?
4. How can logarithms simplify exponents or radicals?
5. How do you solve equations involving logarithms?

Tip: The power rule is particularly useful for simplifying roots or exponents inside a logarithm. Always rewrite roots as fractional exponents!