If `2^(x).9^(2x+3) = 7^(x+5)`, then x =

Options

A. `(5"log"7 +6"log"3)/("log" 162-"log"7)`
B. `(5"log"7-6"log"3)/("log"162 + "log" 7)`
C. `(5"log" 7-6 "log" 3)/("log"162-"log"7)`
D. none of these

Answer

To solve the equation \( 2^x \cdot 9^{2x+3} = 7^{x+5} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 2^x \cdot 9^{2x+3} = 7^{x+5} \]
We know that \( 9 \) can be expressed as \( 3^2 \). Thus, we can rewrite \( 9^{2x+3} \) as:
\[ 9^{2x+3} = (3^2)^{2x+3} = 3^{4x + 6} \]
So, the equation becomes:
\[ 2^x \cdot 3^{4x + 6} = 7^{x + 5} \]

### Step 2: Separate the terms
Now, we can express the equation in terms of \( x \):
\[ 2^x \cdot 3^{4x + 6} = 7^x \cdot 7^5 \]
This simplifies to:
\[ 2^x \cdot 3^{4x + 6} = 7^x \cdot 16807 \quad (\text{since } 7^5 = 16807) \]

### Step 3: Isolate the terms with \( x \)
Next, we can rearrange the equation:
\[ \frac{2^x \cdot 3^{4x + 6}}{7^x} = 16807 \]
This can be rewritten as:
\[ \left(\frac{2 \cdot 3^4}{7}\right)^x \cdot 3^6 = 16807 \]
Calculating \( 3^4 \) gives \( 81 \), so:
\[ \frac{2 \cdot 81}{7} = \frac{162}{7} \]
Thus, we have:
\[ \left(\frac{162}{7}\right)^x \cdot 729 = 16807 \]
(where \( 3^6 = 729 \))

### Step 4: Further simplify
Now we can isolate \( x \):
\[ \left(\frac{162}{7}\right)^x = \frac{16807}{729} \]

### Step 5: Take logarithm of both sides
Taking logarithm on both sides:
\[ \log\left(\left(\frac{162}{7}\right)^x\right) = \log\left(\frac{16807}{729}\right) \]
Using the property of logarithms \( \log(a^b) = b \cdot \log(a) \):
\[ x \cdot \log\left(\frac{162}{7}\right) = \log(16807) - \log(729) \]

### Step 6: Solve for \( x \)
Now we can solve for \( x \):
\[ x = \frac{\log(16807) - \log(729)}{\log\left(\frac{162}{7}\right)} \]

### Step 7: Calculate the logarithms
We know:
- \( \log(16807) = 5 \cdot \log(7) \)
- \( \log(729) = 6 \cdot \log(3) \)

Substituting these values:
\[ x = \frac{5 \cdot \log(7) - 6 \cdot \log(3)}{\log(162) - \log(7)} \]
Where \( \log(162) = \log(2 \cdot 81) = \log(2) + 4 \cdot \log(3) \).

### Final Result
Thus, the value of \( x \) is:
\[ x = \frac{5 \cdot \log(7) - 6 \cdot \log(3)}{\log(2) + 4 \cdot \log(3) - \log(7)} \]