To expand the logarithmic expression $\log \frac{\sqrt[3]{x^2}}{yz^4}$ fully using the properties of logarithms, we proceed as follows:

Step-by-step Expansion:

1. Apply the logarithmic quotient rule: [ \log \frac{\sqrt[3]{x^2}}{yz^4} = \log \sqrt[3]{x^2} - \log(yz^4) ]

2. Expand the numerator using the power and root properties: The cube root can be expressed as a fractional exponent: $\sqrt[3]{x^2} = x^{2/3}.$ So, $\log \sqrt[3]{x^2} = \log(x^{2/3}) = \frac{2}{3} \log x$.

3. Expand the denominator using the product rule: $\log(yz^4) = \log y + \log z^4.$

4. Simplify $\log z^4$ using the power rule: $\log z^4 = 4 \log z.$

Final Expression:

Combining all the terms, we get: $\log \frac{\sqrt[3]{x^2}}{yz^4} = \frac{2}{3} \log x - (\log y + 4 \log z).$

Simplify further: $\log \frac{\sqrt[3]{x^2}}{yz^4} = \frac{2}{3} \log x - \log y - 4 \log z.$

Would you like more details or clarifications?

Related Questions:

1. What are the general rules for expanding logarithms?
2. How do fractional exponents affect logarithmic expressions?
3. What is the difference between the power rule and the product rule in logarithms?
4. How does the logarithmic quotient rule simplify fractions within logarithms?
5. Can this expansion be applied to natural logarithms ($\ln$) as well?

Tip:

When expanding logarithms, always handle roots and exponents first, as they affect individual terms directly.