Let me analyze the mathematical content in the image and help solve the questions presented. Please give me a moment.Here is a detailed breakdown of the questions from the uploaded image and their solutions:

1. Simplification of Equivalent Terms (Task 8)

• Given term: $\frac{4}{x^7}$
• Options to check equivalence:
  • A: $4 - x^7$
  • B: $\frac{1}{x^7}$
  • C: $4x^{-7}$
  • D: $(4 - x)^7$
  • E: $\frac{4}{x^7}$

Solution:

• Start simplifying the options:
  • $B: \frac{1}{x^7}$ does not match because it lacks the numerator $4$.
  • $C: 4x^{-7}$ is equivalent because $x^{-7} = \frac{1}{x^7}$, so $4x^{-7} = \frac{4}{x^7}$.
  • $E: \frac{4}{x^7}$ is exactly the same as the given term.

Correct answers:

• C ($4x^{-7}$)
• E ($\frac{4}{x^7}$)

2. Rational Exponents (Task 9)

• Expression: $\sqrt[3]{x^5}$
• Task: Write as a power.

Solution:

$\sqrt[3]{x^5} = x^{\frac{5}{3}}$

3. Exponential Form to Logarithmic Form (Task 10)

• Given: $3^4 = 81$
• Task: Write in logarithmic form.

Solution:

$3^4 = 81 \implies \log_3{81} = 4$

4. Logarithmic Check (Task 11)

• Given: $\log_{2.2} = 1.92057$
• Task: Show this is valid.

Verification:

$2.2^{1.92057} \approx 4$ This confirms the logarithmic value is correct.

5. Correct Logarithmic Properties (Task 12)

• Statements to check:
  • A: $\log_a{b} = x \implies a^x = b$
  • B: $\log_5{\sqrt{5}} = -x$
  • C: $\log_h{0} = 1$
  • D: $\log_e{f} = e^f = f$
  • E: $\log_g{g} = 1$

Analysis:

• A: True by definition of a logarithm.
• B: Incorrect, $\log_5{\sqrt{5}} = \frac{1}{2}$, not $-x$.
• C: False, $\log_h{0}$ is undefined.
• D: Incorrect interpretation; $\log_e{f} = x \implies e^x = f$, not $e^f = f$.
• E: True, since $\log_g{g^1} = 1$.

Correct answers:

• A, E

6. Logarithmic Expansion (Task 13)

• Expression: $\log{\frac{a}{\sqrt{b} \cdot c}}$

Solution:

Using the logarithmic rules: $\log{\frac{a}{\sqrt{b} \cdot c}} = \log{a} - \log{(\sqrt{b} \cdot c)} = \log{a} - \left(\frac{1}{2}\log{b} + \log{c}\right)$ $= \log{a} - \frac{1}{2}\log{b} - \log{c}$

7. Exponential Equation (Task 14)

• Equation: $0.4^x = 3$
• Task: Solve for $x$ and round to 4 decimals.

Solution:

Take the logarithm of both sides: $\log{(0.4^x)} = \log{3}$ $x \cdot \log{0.4} = \log{3}$ $x = \frac{\log{3}}{\log{0.4}}$

Using a calculator: $x = \frac{\log{3}}{\log{0.4}} \approx -2.7712$

Summary of Results:

1. Task 8: C and E
2. Task 9: $x^{\frac{5}{3}}$
3. Task 10: $\log_3{81} = 4$
4. Task 11: Verified
5. Task 12: A, E
6. Task 13: $\log{a} - \frac{1}{2}\log{b} - \log{c}$
7. Task 14: $x \approx -2.7712$

Would you like further explanations or visualized steps for any specific part?

5 Related Questions:

1. How can logarithmic rules simplify complex fractions like $\log{\frac{a^2}{b \cdot c^3}}$?
2. What is the relationship between exponential and logarithmic equations?
3. Can negative bases in logarithms lead to real solutions?
4. How do rational exponents relate to radical expressions?
5. Why are certain logarithmic values (like $\log_0$) undefined?

Tip:

Always verify logarithmic results by converting them back into their exponential form to check accuracy!