The value of x satisfying `5^logx-3^(logx-1)=3^(logx+1)-5^(logx - 1)` , where the base of logarithm is 10

Explanation

To solve the equation \( 5^{\log x} - 3^{\log x - 1} = 3^{\log x + 1} - 5^{\log x - 1} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 5^{\log x} - 3^{\log x - 1} = 3^{\log x + 1} - 5^{\log x - 1} \]

### Step 2: Simplify the terms
We can rewrite \( 3^{\log x - 1} \) and \( 3^{\log x + 1} \):
\[ 3^{\log x - 1} = \frac{3^{\log x}}{3} \quad \text{and} \quad 3^{\log x + 1} = 3 \cdot 3^{\log x} \]
Substituting these into the equation gives:
\[ 5^{\log x} - \frac{3^{\log x}}{3} = 3 \cdot 3^{\log x} - 5^{\log x - 1} \]

### Step 3: Rewrite \( 5^{\log x - 1} \)
Similarly, we can rewrite \( 5^{\log x - 1} \):
\[ 5^{\log x - 1} = \frac{5^{\log x}}{5} \]
Substituting this into the equation gives:
\[ 5^{\log x} - \frac{3^{\log x}}{3} = 3 \cdot 3^{\log x} - \frac{5^{\log x}}{5} \]

### Step 4: Multiply through by 15 to eliminate fractions
To eliminate the fractions, multiply the entire equation by 15:
\[ 15 \cdot 5^{\log x} - 5 \cdot 3^{\log x} = 45 \cdot 3^{\log x} - 3 \cdot 5^{\log x} \]

### Step 5: Combine like terms
Rearranging the equation gives:
\[ 15 \cdot 5^{\log x} + 3 \cdot 5^{\log x} = 45 \cdot 3^{\log x} + 5 \cdot 3^{\log x} \]
This simplifies to:
\[ 18 \cdot 5^{\log x} = 50 \cdot 3^{\log x} \]

### Step 6: Divide both sides by \( 3^{\log x} \)
Dividing both sides by \( 3^{\log x} \) gives:
\[ \frac{18 \cdot 5^{\log x}}{3^{\log x}} = 50 \]
This can be rewritten using the properties of logarithms:
\[ 18 \cdot \left(\frac{5}{3}\right)^{\log x} = 50 \]

### Step 7: Isolate \( \left(\frac{5}{3}\right)^{\log x} \)
Dividing both sides by 18 gives:
\[ \left(\frac{5}{3}\right)^{\log x} = \frac{50}{18} = \frac{25}{9} \]

### Step 8: Take logarithm on both sides
Taking logarithm base 10 on both sides:
\[ \log x \cdot \log \left(\frac{5}{3}\right) = \log \left(\frac{25}{9}\right) \]

### Step 9: Solve for \( \log x \)
Now, we can isolate \( \log x \):
\[ \log x = \frac{\log \left(\frac{25}{9}\right)}{\log \left(\frac{5}{3}\right)} \]

### Step 10: Calculate \( x \)
Using properties of logarithms:
\[ \log \left(\frac{25}{9}\right) = \log 25 - \log 9 = 2 \log 5 - 2 \log 3 \]
\[ \log \left(\frac{5}{3}\right) = \log 5 - \log 3 \]
Thus:
\[ \log x = \frac{2 \log 5 - 2 \log 3}{\log 5 - \log 3} = 2 \]
So, \( x = 10^2 = 100 \).

### Final Answer
The value of \( x \) is:
\[ \boxed{100} \]