Let’s analyze and solve the two questions in your uploaded image.

Question 7:

Given: log 3 = 0.477
Find the correct statement about log 81.

Solution:

We know: $81 = 3^4$ Using the logarithmic property: $\log(3^4) = 4 \cdot \log(3)$ Substituting $\log(3) = 0.477$: $\log(81) = 4 \cdot 0.477 = 1.908$

Checking Options:

• (a) $\log 81 = 0.477 \times 0.477 \times 0.477 \times 0.477$: Incorrect, since $\log$ does not behave this way.
• (b) $\log 81 = 4(0.477)$: Correct, as shown in the solution above.
• (c) None of the options: Incorrect.
• (d) $\log 81 = 3(0.477)$: Incorrect.

Answer: (b)

Question 8:

Given: $f(x) = \frac{3x + 2}{5}$
Find the inverse, $f^{-1}(x)$.

Solution:

To find the inverse, solve for $x$ in terms of $y$:

1. Let $y = f(x) = \frac{3x + 2}{5}$. $y = \frac{3x + 2}{5}$
2. Multiply through by 5: $5y = 3x + 2$
3. Rearrange to solve for $x$: $3x = 5y - 2$ $x = \frac{5y - 2}{3}$ Thus, the inverse function is: $f^{-1}(x) = \frac{5x - 2}{3}$

Checking Options:

• (a) $(3 - 5x)/2$: Incorrect.
• (b) None of the options: Incorrect.
• (c) $(5x - 2)/3$: Correct, as derived.
• (d) $(5x + 2)/3$: Incorrect.

Answer: (c)

Do you want me to elaborate on any step or property used in solving these questions?

Relative Questions:

1. What are the key logarithmic properties used in such problems?
2. How do you derive the inverse of a linear function in general?
3. Why is $\log(3^4)$ simplified to $4 \cdot \log(3)$?
4. Can you explain the difference between direct and inverse functions?
5. How can you verify an inverse function is correct?

Tip:

Always check your calculations by substituting the inverse back into the original function!