If `(log)_3y=xa n d(log)_2z=x ,` find `72^x` in terms of `ya n dzdot`

Options

A. `yz^(3)`
B. `y^(2)z^(3)`
C. `y^(3)z^(2)`
D. `y^(3)z^(3)`

Explanation

To solve the problem of finding \( 72^x \) in terms of \( y \) and \( z \), we can follow these steps:

### Step 1: Express \( 72 \) in terms of its prime factors
We can express \( 72 \) as:
\[ 72 = 9 \times 8 \]
This can also be written in terms of its prime factors:
\[ 72 = 3^2 \times 2^3 \]

### Step 2: Write \( 72^x \) using the prime factorization
Now, we can express \( 72^x \) as:
\[ 72^x = (3^2 \times 2^3)^x = 3^{2x} \times 2^{3x} \]

### Step 3: Use the given logarithmic equations
We have the equations:
\[ \log_3 y = x \quad \text{and} \quad \log_2 z = x \]
From these, we can express \( y \) and \( z \) in terms of \( x \):
\[ y = 3^x \quad \text{and} \quad z = 2^x \]

### Step 4: Substitute \( 3^{2x} \) and \( 2^{3x} \) in terms of \( y \) and \( z \)
Now, we can express \( 3^{2x} \) and \( 2^{3x} \):
\[ 3^{2x} = (3^x)^2 = y^2 \]
\[ 2^{3x} = (2^x)^3 = z^3 \]

### Step 5: Combine the expressions
Substituting these back into the expression for \( 72^x \):
\[ 72^x = 3^{2x} \times 2^{3x} = y^2 \times z^3 \]

### Final Result
Thus, we have:
\[ 72^x = y^2 z^3 \]