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Let's solve each problem step-by-step.

### 3.
**(a) \(\log_5 125\)**

\[125 = 5^3 \]
\[ \log_5 125 = \log_5 (5^3) = 3 \]

**(b) \(\log_5 \left(\frac{1}{125}\right)\)**

\[ \frac{1}{125} = 5^{-3} \]
\[ \log_5 \left(\frac{1}{125}\right) = \log_5 (5^{-3}) = -3 \]

### 4.
**(a) \(\ln\left(\frac{1}{e}\right)\)**

\[ \frac{1}{e} = e^{-1} \]
\[ \ln\left(\frac{1}{e}\right) = \ln(e^{-1}) = -1 \]

**(b) \(\log_{10} \sqrt{10}\)**

\[ \sqrt{10} = 10^{1/2} \]
\[ \log_{10} \sqrt{10} = \log_{10} (10^{1/2}) = \frac{1}{2} \]

### 5.
**(a) \(e^{4.5}\)**

\[ e^{4.5} \text{ is already in its simplest form.} \]

**(b) \(\log_{10} 0.0001\)**

\[ 0.0001 = 10^{-4} \]
\[ \log_{10} 0.0001 = \log_{10} (10^{-4}) = -4 \]

### 6.
**(a) \(\log_5 2.25\)**

\[ 2.25 = \left(\frac{3}{2}\right)^2 \]
\[ \log_5 2.25 = \log_5 \left(\left(\frac{3}{2}\right)^2\right) = 2 \log_5 \left(\frac{3}{2}\right) \]

**(b) \(\log_4 500\)**

\[ \log_4 500 = \frac{\log_{10} 500}{\log_{10} 4} \]

### 7.
**(a) \(\log_6 6 - \log_6 15 + \log_6 20\)**

\[ \log_6 6 = 1 \]
\[ \log_6 15 = \log_6 (3 \times 5) = \log_6 3 + \log_6 5 \]
\[ \log_6 20 = \log_6 (2^2 \times 5) = 2 \log_6 2 + \log_6 5 \]

Combining them:
\[ 1 - (\log_6 3 + \log_6 5) + (2 \log_6 2 + \log_6 5) = 1 - \log_6 3 + 2 \log_6 2 \]

**(b) \(\log_{100} 100 - \log_{100} 18 - \log_{100} 50\)**

\[ \log_{100} 100 = 1 \]
\[ \log_{100} 18 + \log_{100} 50 = \log_{100} (18 \times 50) \]
\[ \log_{100} 900 = \log_{100} (100 \times 9) = \log_{100} 100 + \log_{100} 9 = 1 + \log_{100} 9 \]

Combining them:
\[ 1 - (1 + \log_{100} 9) = - \log_{100} 9 \]

### 8.
**(a) \(e^{2 + \ln 5}\)**

\[ e^{2 + \ln 5} = e^2 \times e^{\ln 5} = e^2 \times 5 \]

**(b) \(\ln \left(e^{e^4}\right)\)**

\[ \ln \left(e^{e^4}\right) = e^4 \]

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Would you like any further details or have any questions on any of these solutions? 

Here are 8 related questions to expand on the topic:

1. How would you express \(\log_2 32\) in its simplest form?
2. Can you simplify \(\ln(e^7)\)?
3. What is the value of \(\log_3 \frac{27}{81}\)?
4. How do you solve \(\log_7 49\)?
5. What is the result of \(\ln(\frac{1}{e^3})\)?
6. How do you simplify \(\log_{10} 1000\)?
7. Can you calculate \(\log_8 64\)?
8. How would you solve \(\log_2 0.5\)?

**Tip:** Always remember that logarithms and exponentials are inverse operations. This is key to simplifying many expressions.

Explore how to solve logarithmic and exponential equations with detailed step-by-step solutions. This includes logarithms to different bases, natural logarithms, and common logarithms. Suitable for high school and above.

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