Condense each expression to a single logarithm. $$ \begin{array}{ll} ext { 13) } \log 3-\log 8 & ext { 14) } \frac{\log 6}{3} \ ext { 15) } 4 \log 3-4 \log 8 & ext { 16) } \log 2+\log 11+\log 7 \ ext { 17) } \log _{7} 7-2 \log 12 & ext { 18) } \frac{2 \log 7}{3} \ \begin{array}{ll} ext { 19) } 6 \log _{3} u+6 \log _{3} v & ext { 20) } \ln x-4 \ln y \ ext { 21) } \log _{4} u-6 \log _{4} v & ext { 22) } \log _{3} u-5 \log _{3} v \ ext { 23) } 20 \log _{6} u+5 \log _{6} v & ext { 24) } 4 \log _{3} u-20 \log _{3} v\end{array}\end{array} . \begin{array}{l} ext { (1) }\end{array} $$

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Condense each expression to a single logarithm. $$ \begin{array}{ll}	ext { 13) } \log 3-\log 8 & 	ext { 14) } \frac{\log 6}{3} \ 	ext { 15) } 4 \log 3-4 \log 8 & 	ext { 16) } \log 2+\log 11+\log 7 \ 	ext { 17) } \log _{7} 7-2 \log 12 & 	ext { 18) } \frac{2 \log 7}{3} \ \begin{array}{ll}	ext { 19) } 6 \log _{3} u+6 \log _{3} v & 	ext { 20) } \ln x-4 \ln y \ 	ext { 21) } \log _{4} u-6 \log _{4} v & 	ext { 22) } \log _{3} u-5 \log _{3} v \ 	ext { 23) } 20 \log _{6} u+5 \log _{6} v & 	ext { 24) } 4 \log _{3} u-20 \log _{3} v\end{array}\end{array} . \begin{array}{l}	ext { (1) }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Below are the condensed forms of each logarithmic expression reduced to a single logarithm:

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**13) $$\log 3 - \log 8$$**

Using the quotient rule:
$$
\log \left(\frac{3}{8}\right)
$$

---

**14) $$\frac{\log 6}{3}$$**

Using the power rule:
$$
\log \left(6^{\frac{1}{3}}\right) \quad \text{or} \quad \log \sqrt[3]{6}
$$

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**15) $$4 \log 3 - 4 \log 8$$**

Factor out the 4 and apply the quotient rule:
$$
4 \left(\log 3 - \log 8\right) = \log \left(\left(\frac{3}{8}\right)^4\right)
$$

---

**16) $$\log 2 + \log 11 + \log 7$$**

Using the product rule:
$$
\log (2 \times 11 \times 7) = \log 154
$$

---

**17) $$\log_{7} 7 - 2 \log_{7} 12$$**

Simplify and apply the quotient rule:
$$
\log_{7} 7 - \log_{7} (12^2) = \log_{7} \left(\frac{7}{144}\right)
$$

---

**18) $$\frac{2 \log 7}{3}$$**

Using the power and then the root:
$$
\log \left(7^{\frac{2}{3}}\right) \quad \text{or} \quad \log \sqrt[3]{7^2}
$$

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**19) $$6 \log_{3} u + 6 \log_{3} v$$**

Factor out the 6 and apply the product rule:
$$
6 (\log_{3} u + \log_{3} v) = \log_{3} (u \times v)^6 = \log_{3} (u^6 v^6)
$$

---

**20) $$\ln x - 4 \ln y$$**

Using the quotient rule:
$$
\ln \left(\frac{x}{y^4}\right)
$$

---

**21) $$\log_{4} u - 6 \log_{4} v$$**

Using the quotient rule:
$$
\log_{4} \left(\frac{u}{v^6}\right)
$$

---

**22) $$\log_{3} u - 5 \log_{3} v$$**

Using the quotient rule:
$$
\log_{3} \left(\frac{u}{v^5}\right)
$$

---

**23) $$20 \log_{6} u + 5 \log_{6} v$$**

Using the product rule:
$$
20 \log_{6} u + 5 \log_{6} v = \log_{6} (u^{20} \times v^5) = \log_{6} (u^{20} v^5)
$$

---

**24) $$4 \log_{3} u - 20 \log_{3} v$$**

Using the quotient rule:
$$
4 \log_{3} u - 20 \log_{3} v = \log_{3} \left(\frac{u^4}{v^{20}}\right)
$$

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These condensed forms utilize the fundamental logarithmic identities: the product rule ($$\log a + \log b = \log(ab)$$), the quotient rule ($$\log a - \log b = \log(a/b)$$), and the power rule ($$k \log a = \log(a^k)$$) to combine multiple logarithmic terms into a single logarithm.
Here are the condensed logarithmic expressions: 13) $$\log \left(\frac{3}{8}\right)$$ 14) $$\log \sqrt[3]{6}$$ 15) $$\log \left(\frac{3}{8}\right)^4$$ 16) $$\log 154$$ 17) $$\log_{7} \left(\frac{7}{144}\right)$$ 18) $$\log \sqrt[3]{7^2}$$ 19) $$\log_{3} (u^6 v^6)$$ 20) $$\ln \left(\frac{x}{y^4}\right)$$ 21) $$\log_{4} \left(\frac{u}{v^6}\right)$$ 22) $$\log_{3} \left(\frac{u}{v^5}\right)$$ 23) $$\log_{6} (u^{20} v^5)$$ 24) $$\log_{3} \left(\frac{u^4}{v^{20}}\right)$$

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