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Let's solve each expression one by one, showing the work.

### (a) \(\ 10^{\log_{10} 64}\)
By the properties of logarithms:
\[
10^{\log_{10} 64} = 64
\]
So, the answer is:
\[
64
\]

### (b) \(\ \ln(e^3)\)
By the properties of logarithms, specifically the natural logarithm (ln):
\[
\ln(e^3) = 3 \cdot \ln(e) = 3 \cdot 1 = 3
\]
So, the answer is:
\[
3
\]

### (c) \(\ \log_3\left(\sqrt{27}\right)\)
We know that \(\sqrt{27} = 27^{1/2}\), so:
\[
\log_3\left(\sqrt{27}\right) = \log_3\left(27^{1/2}\right) = \frac{1}{2} \cdot \log_3(27)
\]
Since \(27 = 3^3\), we have:
\[
\log_3(27) = \log_3(3^3) = 3
\]
Thus:
\[
\frac{1}{2} \cdot \log_3(27) = \frac{1}{2} \cdot 3 = \frac{3}{2}
\]
So, the answer is:
\[
\frac{3}{2}
\]

### (d) \(\ \log_2(80) - \log_2(5)\)
Using the logarithm subtraction property:
\[
\log_2(80) - \log_2(5) = \log_2\left(\frac{80}{5}\right)
\]
\[
\frac{80}{5} = 16, \quad \text{and} \quad 16 = 2^4
\]
So:
\[
\log_2(16) = \log_2(2^4) = 4
\]
So, the answer is:
\[
4
\]

### (e) \(\ \log_8(4)\)
We can express 4 as a power of 8:
\[
4 = 8^{2/3}
\]
So:
\[
\log_8(4) = \log_8\left(8^{2/3}\right) = \frac{2}{3}
\]
So, the answer is:
\[
\frac{2}{3}
\]

### (f) \(\ \log_{15}(25) + \log_{15}(9)\)
Using the logarithm addition property:
\[
\log_{15}(25) + \log_{15}(9) = \log_{15}(25 \times 9)
\]
\[
25 \times 9 = 225, \quad \text{and} \quad 225 = 15^2
\]
So:
\[
\log_{15}(225) = \log_{15}(15^2) = 2
\]
So, the answer is:
\[
2
\]

### Summary of Answers:
1. (a) \(64\)
2. (b) \(3\)
3. (c) \(\frac{3}{2}\)
4. (d) \(4\)
5. (e) \(\frac{2}{3}\)
6. (f) \(2\)

Would you like any further details or explanations on these solutions?

### Related Questions:
1. How would you solve \(\log_{2}(32) + \log_{2}(8)\)?
2. What is the value of \(\ln(e^x)\) for any \(x\)?
3. How do you solve \(\log_5(125)\)?
4. What is \(\log_{10}(100)\)?
5. How can you simplify \(\log_7(49) - \log_7(7)\)?
6. What is \(\log_4(16)\)?
7. How do you find the value of \(\log_3(81)\)?
8. How do you solve \(\log_9(27)\)?

### Tip:
Remember that logarithms are the inverses of exponentiation, and understanding their properties can greatly simplify complex expressions!

Learn how to solve logarithmic expressions step-by-step with detailed examples. This page covers properties of logarithms, including logarithm addition, subtraction, and simplification. Suitable for high school students and anyone preparing for logarithmic calculations.

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