This is a math worksheet containing logarithmic and exponential function problems. Let’s solve the questions step-by-step for clarity.

1. (0,56) Determine which graph corresponds to $y = \left( \frac{3}{5} \right)^x$.

• The function $y = \left( \frac{3}{5} \right)^x$ is an exponential decay function because $\frac{3}{5} \in (0, 1)$.
• In an exponential decay, the graph decreases as $x$ increases.
• By analyzing the given graphs:
  • Graph A shows exponential decay.
  • The other graphs (B, V, G, D) show exponential growth or other forms.
• Answer: А

2. (0,56) Find $x$ if $5^x = 3$.

To solve for $x$, take the logarithm of both sides:

$x = \log_5 3$

• This matches option Г (x = \log_5 3).
• Answer: Г

3. (0,56) Identify the incorrect inequality.

A. $3^{-1} = \frac{1}{3}$: True.
B. $4^0 = 1$: True.
C. $2^5 = 2 \cdot 2^6$: Incorrect.

• The correct equality is $2^5 = 2 \cdot 2^4$ because $2 \cdot 2^6 = 2^7$.
  D. $\log_7 7 = 1$: True.

• Answer: В

4. (1,5) Match expressions (1-3) to their values (А-Д):

Expressions:

1. $\log_3 2 + \log_3 16$
2. $2 \log_5 3 - \log_5 9$
3. $16^{\frac{3}{4}}$

Solutions:

1. Use the logarithmic property $\log_a m + \log_a n = \log_a (mn)$: $\log_3 2 + \log_3 16 = \log_3 (2 \cdot 16) = \log_3 32$

   • Value: В (log_3 32).

2. Simplify $2 \log_5 3 - \log_5 9$: $\log_5 3^2 - \log_5 9 = \log_5 9 - \log_5 9 = 0$

   • Value: А (0).

3. Simplify $16^{\frac{3}{4}}$:

   • $16 = 2^4$, so: $16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{3} = 8$
   • Value: Б (8).

Final Matching:

1 → В, 2 → А, 3 → Б.

• Answer: 1–В, 2–А, 3–Б

5. (2,6) Solve $2^{3x - 3} = 35^{x - 1}$.

Would you like detailed solutions for Questions 5–8, or just the answers? Let me know!

Related Questions:

1. How can we graphically determine the behavior of an exponential decay function?
2. What logarithmic properties help simplify expressions like $\log_a m + \log_a n$?
3. How do we solve equations of the form $a^x = b$?
4. Why is $2^5 = 2 \cdot 2^4$ but not $2 \cdot 2^6$?
5. How do we find the domain of a logarithmic function?

Tip: Remember, for $y = a^x$, if $a \in (0,1)$, the graph decreases (exponential decay). If $a > 1$, it increases (exponential growth).