To condense the given expression to a single logarithm, we use the properties of logarithms:

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: $k \cdot \log(a) = \log(a^k)$.

The given expression is:

$\log(x) - 12\log(y) + 7\log(z).$

Step 1: Apply the power rule

Rewrite $12\log(y)$ as $\log(y^{12})$ and $7\log(z)$ as $\log(z^7)$:

$\log(x) - \log(y^{12}) + \log(z^7).$

Step 2: Combine terms using the product and quotient rules

First, combine $\log(x)$ and $\log(z^7)$ using the product rule:

$\log(x) + \log(z^7) = \log(x \cdot z^7).$

Then subtract $\log(y^{12})$ using the quotient rule:

$\log(x \cdot z^7) - \log(y^{12}) = \log\left(\frac{x \cdot z^7}{y^{12}}\right).$

Final Answer:

$\log\left(\frac{x \cdot z^7}{y^{12}}\right).$

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Would you like a step-by-step breakdown of any part of the process?

5 Related Questions:

1. What are the properties of logarithms and when are they used?
2. How do you expand logarithmic expressions?
3. What is the difference between natural logarithms and common logarithms?
4. How would you handle an expression like $2\log(a) - 3\log(b) + \log(c)$?
5. What happens if there are negative numbers inside a logarithm?

Tip:

Always check that the arguments of logarithms are positive before simplifying, as logarithms are undefined for non-positive values.