If x + `log_(15) (1 + 3^(x))= x log_(15) 5 + log_(15) 12`, where x is an integer. Then what is x equal to ?

Options

A. -3
B. 2
C. 1
D. 3

Answer

To solve the equation \( x + \log_{15}(1 + 3^x) = x \log_{15}(5) + \log_{15}(12) \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x + \log_{15}(1 + 3^x) = x \log_{15}(5) + \log_{15}(12) \]

### Step 2: Use logarithmic properties
We can rewrite \( x \) as \( \log_{15}(15^x) \) because \( \log_{a}(a^b) = b \):
\[ \log_{15}(15^x) + \log_{15}(1 + 3^x) = x \log_{15}(5) + \log_{15}(12) \]

### Step 3: Combine the logarithms on the left side
Using the property \( \log_{a}(m) + \log_{a}(n) = \log_{a}(mn) \):
\[ \log_{15}(15^x (1 + 3^x)) = x \log_{15}(5) + \log_{15}(12) \]

### Step 4: Rewrite the right side
We can also combine the right side using the same logarithmic property:
\[ \log_{15}(15^x (1 + 3^x)) = \log_{15}(5^x \cdot 12) \]

### Step 5: Set the arguments equal
Since the logarithms are equal, we can set the arguments equal to each other:
\[ 15^x (1 + 3^x) = 5^x \cdot 12 \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ 15^x + 15^x \cdot 3^x = 5^x \cdot 12 \]

### Step 7: Factor out common terms
Notice that \( 15^x = (3 \cdot 5)^x = 3^x \cdot 5^x \):
\[ 3^x \cdot 5^x + 3^x \cdot 5^x \cdot 3^x = 5^x \cdot 12 \]
\[ 3^x \cdot 5^x (1 + 3^x) = 5^x \cdot 12 \]

### Step 8: Divide both sides by \( 5^x \)
Assuming \( 5^x \neq 0 \):
\[ 3^x (1 + 3^x) = 12 \]

### Step 9: Let \( y = 3^x \)
Substituting \( y = 3^x \) gives us:
\[ y(1 + y) = 12 \]
\[ y^2 + y - 12 = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{-1 \pm \sqrt{1 + 48}}{2} = \frac{-1 \pm 7}{2} \]
This gives us:
\[ y = 3 \quad \text{or} \quad y = -4 \]

Since \( y = 3^x \) must be positive, we take \( y = 3 \):
\[ 3^x = 3 \implies x = 1 \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{1} \]