Solve for x :
3^(log x)-2^(log x) =2^(log x+1)-3^(log x-1)

Answer

To solve the equation \( 3^{(\log x)} - 2^{(\log x)} = 2^{(\log x + 1)} - 3^{(\log x - 1)} \), we can follow these steps:

### Step 1: Substitute \( \log x \) with a variable
Let \( t = \log x \). Then, the equation becomes:
\[ 3^t - 2^t = 2^{(t + 1)} - 3^{(t - 1)} \]

### Step 2: Simplify the right-hand side
We can rewrite \( 2^{(t + 1)} \) and \( 3^{(t - 1)} \):
\[ 2^{(t + 1)} = 2^t \cdot 2 \]
\[ 3^{(t - 1)} = \frac{3^t}{3} \]
Now, substituting these into the equation gives:
\[ 3^t - 2^t = 2 \cdot 2^t - \frac{3^t}{3} \]

### Step 3: Rearrange the equation
Multiply through by 3 to eliminate the fraction:
\[ 3(3^t - 2^t) = 3(2 \cdot 2^t) - 3^t \]
This simplifies to:
\[ 3^{t + 1} - 3 \cdot 2^t = 6 \cdot 2^t - 3^t \]
Now, rearranging gives:
\[ 3^{t + 1} + 3^t = 9 \cdot 2^t \]

### Step 4: Factor out common terms
Notice that \( 3^t \) can be factored out:
\[ 3^t(3 + 1) = 9 \cdot 2^t \]
This simplifies to:
\[ 4 \cdot 3^t = 9 \cdot 2^t \]

### Step 5: Divide both sides by \( 2^t \)
\[ \frac{4 \cdot 3^t}{2^t} = 9 \]
This can be rewritten as:
\[ 4 \left(\frac{3}{2}\right)^t = 9 \]

### Step 6: Isolate \( \left(\frac{3}{2}\right)^t \)
Divide both sides by 4:
\[ \left(\frac{3}{2}\right)^t = \frac{9}{4} \]

### Step 7: Take logarithm on both sides
Taking logarithm (base 10) gives:
\[ t \cdot \log\left(\frac{3}{2}\right) = \log\left(\frac{9}{4}\right) \]

### Step 8: Solve for \( t \)
\[ t = \frac{\log\left(\frac{9}{4}\right)}{\log\left(\frac{3}{2}\right)} \]

### Step 9: Substitute back to find \( x \)
Since \( t = \log x \), we have:
\[ \log x = \frac{\log\left(\frac{9}{4}\right)}{\log\left(\frac{3}{2}\right)} \]
Thus:
\[ x = 10^{\frac{\log\left(\frac{9}{4}\right)}{\log\left(\frac{3}{2}\right)}} \]

### Step 10: Calculate \( x \)
Calculating \( x \):
\[ x = \left(\frac{9}{4}\right)^{\frac{1}{\log\left(\frac{3}{2}\right)}} \]
This simplifies to:
\[ x = 100 \]

### Final Answer:
\[ x = 100 \]