To fully expand the given logarithmic expression using the properties of logarithms, let's break it down step-by-step.

Expression:

$\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right)$

Step 1: Apply the logarithm of a quotient rule

Using the property: $\log \left( \frac{A}{B} \right) = \log A - \log B$ we get: $\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right) = \log \left( x^3 \sqrt{z^5} \right) - \log \left( y^4 \right)$

Step 2: Expand the first logarithm (numerator) using the logarithm of a product

Using the property: $\log(AB) = \log A + \log B$ we can expand: $\log \left( x^3 \sqrt{z^5} \right) = \log (x^3) + \log \left( \sqrt{z^5} \right)$

Step 3: Simplify using the logarithm of a power rule

• For $\log(x^3)$, apply: $\log(x^n) = n \log(x)$ Thus: $\log(x^3) = 3 \log x$

• For $\log \left( \sqrt{z^5} \right)$, rewrite $\sqrt{z^5}$ as $z^{5/2}$. Then apply the power rule: $\log \left( z^{5/2} \right) = \frac{5}{2} \log z$

Step 4: Expand the second logarithm (denominator)

• For $\log(y^4)$, apply the power rule: $\log(y^4) = 4 \log y$

Final Expression:

Now, putting all the pieces together: $\log \left( \frac{x^3 \sqrt{z^5}}{y^4} \right) = 3 \log x + \frac{5}{2} \log z - 4 \log y$

Thus, the fully expanded logarithmic expression is: $3 \log x + \frac{5}{2} \log z - 4 \log y$

Would you like further details on any part of this explanation? Here are a few related questions for practice:

1. How would you expand $\log \left( \frac{a^2 b}{c^3} \right)$?
2. Can you express $\log \left( x^4 y^5 \right)$ using properties of logarithms?
3. What is the simplified form of $\log \left( \sqrt{a^3 b} \right)$?
4. How would you expand $\log \left( \frac{x^2 y}{\sqrt{z}} \right)$?
5. What is the general rule for expanding logarithms of quotients?

Tip: When expanding logarithms, always apply the quotient rule first, followed by the product and power rules!