https://file.aiquickdraw.com/mathbot/file/1727009166276xbnxzl1k.jpg

Let's go through each problem from the image you uploaded:

### Problem 6: 
The property of exponents given is \( a^{\log_b} = b \). The expressions listed need to be simplified based on this rule.

**a.** \( 100^{\log 8} \)

Using the rule:
\[
100^{\log 8} = 8
\]

**b.** \( 32 \)

This is already a simplified constant.

**c.** \( 25^{\frac{5}{8}} + 64^{\frac{1}{5}} \)

Breaking this into steps:
\[
25^{5/8} = (5^2)^{5/8} = 5^{5/4}, \quad 64^{1/5} = 2
\]
So, the result is \( 5^{5/4} + 2 \).

**d.** \( 4^{\frac{\log 7}{8}} = 8^{\frac{2}{\log 2}} + 3^{\frac{\log 5}{8}} \)

This problem requires further simplification by applying the logarithmic properties and exponentiation rules.

### Problem 7: 
This problem involves logarithmic properties.

**a.** \( \frac{1}{2} \log 128 + 2 \log 9 - 27 \log 4 - 6 \log \frac{1}{3} \)

Using properties of logarithms (like \( n \log a = \log a^n \) and \( \log a + \log b = \log (a \cdot b) \)):
- \( \frac{1}{2} \log 128 = \log \sqrt{128} \)
- \( 2 \log 9 = \log 9^2 \)

This can be simplified to a single logarithmic term.

**b.** Similar steps apply to this expression.

### Problem 8:
These logarithmic equations must be converted to exponential form to solve for \( x \).

**a.** \( 2 \log x = 3 \)

Exponential form:
\[
x^2 = 10^3, \quad \therefore x = \sqrt{1000}
\]

**b.** \( 3 \log (2x + 1) = 2 \)

Exponential form:
\[
(2x + 1)^3 = 10^2
\]

**c.** \( \frac{1}{4} \log (6x - 5) = -1 \)

Exponential form:
\[
(6x - 5)^{1/4} = 10^{-1}
\]

Would you like detailed steps for any of these?

Solve logarithmic and exponential problems from the given image, involving properties of exponents and logarithms.

Learn to simplify expressions involving logarithms and exponents, using properties like a^log_b = b. This problem set covers key concepts in algebra, logarithmic properties, and exponentiation rules. Suitable for students in Grades 10-12.

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