If ` 3 +"log"_(5)x = 2"log"_(25) y`, then x equals to

Options

A. `(y)/(125)`
B. `(y)/(25)`
C. `(y^(2))/(25)`
D. `3-(y^(2))/(25)`

Answer

To solve the equation \( 3 + \log_{5} x = 2 \log_{25} y \), we will follow these steps:

### Step 1: Rewrite the Right-Hand Side
We start with the right-hand side, \( 2 \log_{25} y \). We can express \( \log_{25} y \) in terms of base 5:
\[ \log_{25} y = \log_{5} y / \log_{5} 25 \]
Since \( 25 = 5^2 \), we have \( \log_{5} 25 = 2 \). Therefore:
\[ \log_{25} y = \frac{\log_{5} y}{2} \]
Thus,
\[ 2 \log_{25} y = 2 \cdot \frac{\log_{5} y}{2} = \log_{5} y \]

### Step 2: Substitute Back into the Equation
Now we can substitute this back into the original equation:
\[ 3 + \log_{5} x = \log_{5} y \]

### Step 3: Isolate \( \log_{5} x \)
Next, we isolate \( \log_{5} x \) by moving 3 to the right-hand side:
\[ \log_{5} x = \log_{5} y - 3 \]

### Step 4: Rewrite 3 in Logarithmic Form
We can express 3 in terms of logarithms:
\[ 3 = \log_{5} (5^3) = \log_{5} 125 \]
So we can rewrite the equation as:
\[ \log_{5} x = \log_{5} y - \log_{5} 125 \]

### Step 5: Use Logarithmic Properties
Using the property of logarithms that states \( \log_{a} b - \log_{a} c = \log_{a} \left( \frac{b}{c} \right) \), we have:
\[ \log_{5} x = \log_{5} \left( \frac{y}{125} \right) \]

### Step 6: Equate Arguments of Logarithms
Since the logarithms are equal, we can equate their arguments:
\[ x = \frac{y}{125} \]

### Final Result
Thus, the value of \( x \) is:
\[ x = \frac{y}{125} \]

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