We are given the expression:

$\log \sqrt{x} - \log x^2 + \frac{3}{4} \log x^4$

To combine these into a single logarithmic expression, we can use the following logarithmic properties:

1. Power Rule: $\log a^b = b \log a$
2. Quotient Rule: $\log a - \log b = \log \left(\frac{a}{b}\right)$
3. Product Rule: $\log a + \log b = \log (a \cdot b)$

Step 1: Simplify each term.

• The first term is $\log \sqrt{x}$, which can be rewritten as: $\log \sqrt{x} = \log x^{1/2} = \frac{1}{2} \log x$

• The second term is $\log x^2$, which can be rewritten as: $\log x^2 = 2 \log x$

• The third term is $\frac{3}{4} \log x^4$, which can be rewritten as: $\frac{3}{4} \log x^4 = \frac{3}{4} \cdot 4 \log x = 3 \log x$

Step 2: Substitute and combine.

Now, substitute the simplified terms into the original expression:

$\frac{1}{2} \log x - 2 \log x + 3 \log x$

Combine the $\log x$ terms:

$\frac{1}{2} \log x - 2 \log x + 3 \log x = \left(\frac{1}{2} - 2 + 3\right) \log x = \frac{3}{2} \log x$

Step 3: Express as a single logarithm.

Finally, using the power rule, we can rewrite the expression as:

$\frac{3}{2} \log x = \log x^{3/2}$

Thus, the entire expression simplifies to:

$\log x^{3/2}$

Final Answer:

$\log x^{3/2}$

Would you like a detailed explanation of any of the steps? Here are some follow-up questions to expand on this topic:

1. How do the properties of logarithms simplify other types of expressions?
2. What is the difference between the natural logarithm and the common logarithm?
3. How do logarithmic functions relate to exponential functions?
4. Can logarithms with different bases be combined or simplified similarly?
5. How can logarithmic equations be solved algebraically?

Tip: Practice using the logarithmic properties on different types of expressions to build fluency in simplifying and combining logarithmic terms.