https://file.aiquickdraw.com/mathbot/file/17365238672375ottdlnc.jpg

Let's work through each part of the problem and express them as a single logarithm:

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### (a) \( \log 5 + \log 11 \)
Using the **logarithmic property** \( \log a + \log b = \log (a \cdot b) \):
\[
\log 5 + \log 11 = \log (5 \cdot 11) = \log 55
\]

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### (b) \( \log 20 - \log 4 \)
Using the **logarithmic property** \( \log a - \log b = \log \left(\frac{a}{b}\right) \):
\[
\log 20 - \log 4 = \log \left(\frac{20}{4}\right) = \log 5
\]

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### (c) \( \log 6 + \log 8 - \log 12 \)
First, combine \( \log 6 + \log 8 \) using \( \log a + \log b = \log (a \cdot b) \):
\[
\log 6 + \log 8 = \log (6 \cdot 8) = \log 48
\]
Now subtract \( \log 12 \) using \( \log a - \log b = \log \left(\frac{a}{b}\right) \):
\[
\log 48 - \log 12 = \log \left(\frac{48}{12}\right) = \log 4
\]

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### (d) \( 2 \log 3 + 4 \log 2 \)
Using the **logarithmic property** \( n \log a = \log (a^n) \):
\[
2 \log 3 = \log (3^2) = \log 9, \quad 4 \log 2 = \log (2^4) = \log 16
\]
Now combine using \( \log a + \log b = \log (a \cdot b) \):
\[
\log 9 + \log 16 = \log (9 \cdot 16) = \log 144
\]

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### Final Answers:
(a) \( \log 55 \)  
(b) \( \log 5 \)  
(c) \( \log 4 \)  
(d) \( \log 144 \)

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Would you like detailed explanations for any specific part? Let me know!

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Here are 5 related questions for practice:
1. Simplify \( \log 7 + \log 3 - \log 21 \) as a single logarithm.
2. Express \( 3 \log 2 - \log 8 \) as a single logarithm.
3. Combine \( \log 4 + \log 5 + \log 2 \) into a single logarithm.
4. Rewrite \( \frac{1}{2} \log 25 \) as a single logarithmic term.
5. Simplify \( \log 10^3 - 2 \log 5 \) to a single logarithm.

**Tip:** Always check the base of the logarithms when simplifying; the default is base 10 unless otherwise specified.

Express as a single logarithm: (a) log 5 + log 11, (b) log 20 - log 4, (c) log 6 + log 8 - log 12, (d) 2 log 3 + 4 log 2.

Explore step-by-step solutions to express logarithmic expressions as single terms, including examples like log 5 + log 11 and 2 log 3 + 4 log 2. This guide covers the properties of logarithms and is suitable for Grades 10-12.

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