https://file.aiquickdraw.com/mathbot/file/17270090986874j6rlwaf.jpg

Let's go over the questions in the image:

### Question 6:
You are asked to use the property \( a^{\log_b b} = b \) to determine the result of the following expressions:

#### a. \( 100^{\log_8 8} \)
This can be simplified using the property \( a^{\log_b b} = b \), where \( a = 100 \) and \( b = 8 \), so:
\[
100^{\log_8 8} = 100^1 = 100
\]

#### b. \( 32 \)
No need for simplification; the value is directly \( 32 \).

#### c. \( 25^{5\log_8 8} + 64^{8\log_5 5} \)
Again, apply \( a^{\log_b b} = b \) for both terms:
\[
25^{5\log_8 8} = 25^5, \quad 64^{8\log_5 5} = 64^8
\]
You can simplify further by calculating the powers.

#### d. \( 4^{\log_7 8} - 8^{\log_2 2} + 3^{\log_5 \frac{5}{8}} \)
Similarly, use the property to simplify each term, then compute accordingly.

### Question 7:
You are asked to apply logarithmic properties to determine the results of the expressions.

#### a. \( \frac{1}{2} \log 128 + 2 \log 9 - 27 \log 4 - 6 \log \frac{1}{3} \)
This can be simplified using the logarithmic properties, such as:
- \( a \log b = \log b^a \)
- \( \log(ab) = \log a + \log b \)

#### b. \( 3 \log \frac{1}{9} - 2 \log 8 + 2^{8 \log 27} - \frac{1}{9} \log \frac{1}{3} \)
Apply logarithmic properties and simplify step-by-step.

### Question 8:
You are asked to convert the following logarithmic forms to exponential forms and determine the value of \( x \).

#### a. \( 2^{\log x} = 3 \)
Convert to exponential form:
\[
x = 3^2 = 9
\]

#### b. \( 3^{\log(2x + 1)} = 2 \)
Solve for \( x \).

#### c. \( \frac{1}{4} \log(6x - 5) = -1 \)
Solve for \( x \) after converting to exponential form.

Would you like detailed step-by-step solutions to these problems? Here are 5 questions to explore further:

1. How do you apply the logarithmic power rule?
2. What happens if the base and argument of a logarithm don't match?
3. How do you handle complex expressions with mixed logs and powers?
4. Can you explain the logarithmic change of base formula?
5. What are common mistakes when converting between log and exponential forms?

**Tip:** Always double-check your base and exponent in logarithmic equations to avoid simple errors!

Use logarithmic properties to determine the result of the given expressions, including simplifying logarithms and converting to exponential form.

Learn how to solve logarithmic expressions using properties like a^{log_b b} = b, applying the power and product rules for logarithms, and converting logarithmic forms to exponential forms. Suitable for high school students (Grades 10-12).

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