If `log_(5) x = y`, find `5^(2y + 3)` in terms of x.

Answer

To solve the problem where \( \log_{5} x = y \) and we need to find \( 5^{2y + 3} \) in terms of \( x \), we can follow these steps:

### Step 1: Start with the given logarithmic equation
We have:
\[ \log_{5} x = y \]

### Step 2: Convert the logarithmic equation to exponential form
Using the definition of logarithms, we can rewrite the equation in exponential form:
\[ x = 5^y \]

### Step 3: Multiply both sides of the logarithmic equation by 2
To find \( 5^{2y} \), we multiply both sides of the equation \( \log_{5} x = y \) by 2:
\[ 2 \log_{5} x = 2y \]

### Step 4: Use the power rule of logarithms
Using the power rule of logarithms, we can rewrite the left side:
\[ \log_{5} (x^2) = 2y \]

### Step 5: Convert this new logarithmic equation to exponential form
Now, we convert this back to exponential form:
\[ x^2 = 5^{2y} \]

### Step 6: Express \( 5^{2y} \) in terms of \( x \)
From the previous step, we have:
\[ 5^{2y} = x^2 \]

### Step 7: Now, we need to find \( 5^{2y + 3} \)
We can express \( 5^{2y + 3} \) as:
\[ 5^{2y + 3} = 5^{2y} \cdot 5^3 \]

### Step 8: Substitute \( 5^{2y} \) with \( x^2 \)
Substituting \( 5^{2y} \) from Step 6:
\[ 5^{2y + 3} = x^2 \cdot 5^3 \]

### Step 9: Calculate \( 5^3 \)
Since \( 5^3 = 125 \), we can write:
\[ 5^{2y + 3} = 125 x^2 \]

### Final Answer
Thus, we have:
\[ 5^{2y + 3} = 125 x^2 \]