https://file.aiquickdraw.com/mathbot/file/1727227528185l1snp8ix.jpg

Let's solve both parts (a) and (b) from the image using logarithmic properties and then verify the results using the change-of-base formula.

### Part (a):  
\[ \log_2(8) \]

We know that \( 8 = 2^3 \), so:
\[
\log_2(8) = \log_2(2^3) = 3
\]
This is because \( \log_b(b^x) = x \), where \( b \) is the base and \( x \) is the exponent.

#### Verification (Change-of-base formula):
The change-of-base formula is:
\[
\log_b(a) = \frac{\log(a)}{\log(b)}
\]
For \( \log_2(8) \):
\[
\log_2(8) = \frac{\log(8)}{\log(2)}
\]
Since \( \log(8) \approx 0.9031 \) and \( \log(2) \approx 0.3010 \):
\[
\frac{0.9031}{0.3010} \approx 3
\]
Thus, \( \log_2(8) = 3 \) is correct.

### Part (b):
\[ \log_5 \left( \frac{1}{5} \right) \]

We can express \( \frac{1}{5} \) as \( 5^{-1} \), so:
\[
\log_5 \left( \frac{1}{5} \right) = \log_5 \left( 5^{-1} \right) = -1
\]
This follows from the same property \( \log_b(b^x) = x \).

#### Verification (Change-of-base formula):
Using the change-of-base formula again:
\[
\log_5 \left( \frac{1}{5} \right) = \frac{\log \left( \frac{1}{5} \right)}{\log(5)}
\]
We know that \( \log \left( \frac{1}{5} \right) = -\log(5) \), so:
\[
\frac{-\log(5)}{\log(5)} = -1
\]
Thus, \( \log_5 \left( \frac{1}{5} \right) = -1 \) is correct.

### Final Answers:
- (a) \( \log_2(8) = 3 \)
- (b) \( \log_5 \left( \frac{1}{5} \right) = -1 \)

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

Here are some related questions for practice:
1. Simplify \( \log_2(16) \).
2. Solve \( \log_3(27) \).
3. Simplify \( \log_7 \left( \frac{1}{49} \right) \).
4. Use the change-of-base formula to compute \( \log_4(64) \).
5. Solve \( \log_9 \left( \frac{1}{81} \right) \).

**Tip**: Remember, if the base and the argument of the logarithm are the same (like \( \log_b(b^x) \)), the answer is always the exponent \( x \).

Use properties of logarithms or a definition to simplify each expression. Check each result with a change-of-base formula. (a) log₂(8), (b) log₅(1/5)

Learn how to simplify logarithmic expressions using logarithmic properties and the change-of-base formula. This example involves simplifying log₂(8) and log₅(1/5) and verifying the results using algebraic methods. Suitable for high school students in Grades 9-12.

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