To rewrite each expression as a single logarithm, we can apply the properties of logarithms:

1. Logarithmic subtraction property: $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$
2. Logarithmic addition property: $\log_b(a) + \log_b(c) = \log_b(ac)$
3. Power property: $n \cdot \log_b(a) = \log_b(a^n)$

Using these properties, we can simplify each expression:

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1. Given: $\log_5(x) - \log_5(y)$

   Solution: $\log_5\left(\frac{x}{y}\right)$

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2. Given: $3 \log_8(x) + \log_8(y)$

   Solution: First, apply the power property to $3 \log_8(x)$: $3 \log_8(x) = \log_8(x^3)$ Then, use the addition property: $\log_8(x^3) + \log_8(y) = \log_8(x^3 \cdot y)$

   Final answer: $\log_8(x^3 \cdot y)$

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3. Given: $\ln(x) + \ln(y) - \ln(z)$

   Solution: First, use the addition property for $\ln(x) + \ln(y)$: $\ln(x) + \ln(y) = \ln(x \cdot y)$ Then, apply the subtraction property: $\ln(x \cdot y) - \ln(z) = \ln\left(\frac{x \cdot y}{z}\right)$

   Final answer: $\ln\left(\frac{x \cdot y}{z}\right)$

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Let me know if you have any questions, or if you'd like a deeper explanation on any of the logarithmic properties used here.

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Here are 5 related questions to deepen understanding:

1. How would you rewrite $2 \log_7(x) - \log_7(y) + \log_7(z)$ as a single logarithm?
2. Can you simplify $\ln(a) - 2\ln(b) + \ln(c)$ to a single natural logarithm?
3. What does $\log_3(a) + \log_3(b) - 4\log_3(c)$ simplify to in a single logarithmic expression?
4. If you have $\log(x^2)$ and need to express it without an exponent, how would you simplify it?
5. What is the simplified form of $5 \log(x) - \log(y)$ using properties of logarithms?

Tip: Remember, whenever you see multiple logarithmic terms with the same base, try using the properties of logarithms to combine them into a single expression.