4. Two experiments revealed the relationship between variables x and y is described by 3(27^x) = 81^y and log3 y = 2 + log3(x-2). Calculate the values of x and y

Let's solve the given problem step-by-step.

We are given the following two equations:

1. \( 3(27^x) = 81^y \)
2. \( \log_3 y = 2 + \log_3(x - 2) \)

### Step 1: Simplify the first equation

First, let's express \( 27 \) and \( 81 \) in terms of base \( 3 \):

\[ 27 = 3^3 \]
\[ 81 = 3^4 \]

Substitute these into the first equation:

\[ 3(3^{3x}) = (3^4)^y \]

This simplifies to:

\[ 3^{1 + 3x} = 3^{4y} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 1 + 3x = 4y \]

### Step 2: Solve for \( y \) in terms of \( x \)

\[ 4y = 1 + 3x \]
\[ y = \frac{1 + 3x}{4} \]

### Step 3: Simplify the second equation

Using the logarithm properties, we can rewrite the second equation:

\[ \log_3 y = 2 + \log_3(x - 2) \]

Apply the property \(\log_b(m) + \log_b(n) = \log_b(mn)\):

\[ \log_3 y = \log_3 3^2 + \log_3(x - 2) \]

\[ \log_3 y = \log_3(9) + \log_3(x - 2) \]

Combine the logarithms:

\[ \log_3 y = \log_3[9(x - 2)] \]

Since the bases are the same, we can equate the arguments:

\[ y = 9(x - 2) \]

### Step 4: Substitute \( y \) from Step 2 into the equation from Step 3

\[ \frac{1 + 3x}{4} = 9(x - 2) \]

### Step 5: Solve for \( x \)

Multiply both sides by 4 to clear the fraction:

\[ 1 + 3x = 36(x - 2) \]

Distribute on the right side:

\[ 1 + 3x = 36x - 72 \]

Combine like terms:

\[ 1 + 72 = 36x - 3x \]

\[ 73 = 33x \]

\[ x = \frac{73}{33} \]

### Step 6: Solve for \( y \)

Using the value of \( x \):

\[ y = \frac{1 + 3x}{4} \]

\[ y = \frac{1 + 3 \left(\frac{73}{33}\right)}{4} \]

\[ y = \frac{1 + \frac{219}{33}}{4} \]

\[ y = \frac{\frac{33}{33} + \frac{219}{33}}{4} \]

\[ y = \frac{\frac{252}{33}}{4} \]

\[ y = \frac{252}{132} \]

\[ y = \frac{21}{11} \]

### Final Values:

\[ x = \frac{73}{33} \approx 2.21 \]
\[ y = \frac{21}{11} \approx 1.91 \]

If you have any questions or want further details, please let me know!

Here are some related questions for further practice:

1. What are the properties of logarithms that are useful for solving equations?
2. How can you simplify exponential equations?
3. What are the steps for converting between exponential and logarithmic forms?
4. How do you solve logarithmic equations with different bases?
5. Can you solve for \( x \) in the equation \( 2^{2x+1} = 32 \)?
6. What is the change of base formula for logarithms?
7. How do you solve an equation involving a product of logarithms?
8. How can you check your solutions for logarithmic and exponential equations?

**Tip:** Always ensure to express all terms in an equation to the same base when dealing with exponents and logarithms. This simplifies solving the equations significantly.

Learn how to solve complex exponential and logarithmic equations using examples such as 3(27^x) = 81^y and log3 y = 2 + log3(x-2). This problem involves advanced concepts in algebra, including exponential equations with different bases and logarithmic properties. Suitable for advanced high school students and those preparing for mathematical competitions.

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