Let's solve the problem shown in the image, which asks to condense the given logarithmic expression into a single logarithm using the properties of logarithms.

The expression provided is:

$\log(x) - \frac{1}{2} \log(y) + 3 \log(z)$

Step-by-Step Solution:

We can use the following logarithmic properties:

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

Apply the Power Rule:

• $\frac{1}{2} \log(y)$ becomes $\log(y^{1/2}) = \log(\sqrt{y})$
• $3 \log(z)$ becomes $\log(z^3)$

Now, the expression becomes:

$\log(x) - \log(\sqrt{y}) + \log(z^3)$

Apply the Quotient Rule:

• Combine the first two terms using the quotient rule: $\log\left(\frac{x}{\sqrt{y}}\right)$

Apply the Product Rule:

• Now, combine the result with $\log(z^3)$: $\log\left(\frac{x}{\sqrt{y}} \cdot z^3\right)$

Final Expression:

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

Thus, the condensed expression is:

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

Let me know if you need further details or clarifications!

Related Questions:

1. How does the power rule simplify logarithmic expressions?
2. What are the differences between the product rule and the quotient rule of logarithms?
3. How can logarithmic properties be used to expand logarithmic expressions instead of condensing them?
4. How does the behavior of logarithms change when their base changes?
5. How do we deal with logarithmic expressions involving more complex exponents?

Tip:

Always simplify powers and roots before applying the product and quotient rules in logarithms—it makes the combination of terms easier!