Solve : `7^(log x)=98-x^(log 7)`

Answer

To solve the equation \( 7^{\log x} = 98 - x^{\log 7} \), we can follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We start with the equation:
\[ 7^{\log x} = 98 - x^{\log 7} \]
Using the property of logarithms, \( a^{\log_b c} = c^{\log_b a} \), we can rewrite \( 7^{\log x} \) as \( x^{\log 7} \):
\[ x^{\log 7} = 98 - x^{\log 7} \]

### Step 2: Combine like terms
Now, we can add \( x^{\log 7} \) to both sides of the equation:
\[ x^{\log 7} + x^{\log 7} = 98 \]
This simplifies to:
\[ 2x^{\log 7} = 98 \]

### Step 3: Isolate \( x^{\log 7} \)
Next, we divide both sides by 2:
\[ x^{\log 7} = \frac{98}{2} = 49 \]

### Step 4: Rewrite 49 as a power of 7
We know that \( 49 = 7^2 \), so we can rewrite the equation:
\[ x^{\log 7} = 7^2 \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ \log 7 \cdot \log x = 2 \]

### Step 6: Solve for \( \log x \)
Now, we can isolate \( \log x \):
\[ \log x = \frac{2}{\log 7} \]

### Step 7: Convert back to exponential form
To find \( x \), we convert from logarithmic form to exponential form:
\[ x = 10^{\frac{2}{\log 7}} \]

### Step 8: Calculate \( x \)
Using the change of base formula, we can calculate \( \log 7 \) (approximately 0.845):
\[ x \approx 10^{\frac{2}{0.845}} \approx 10^{2.36} \approx 229.08 \]

### Final Answer
Thus, the solution to the equation \( 7^{\log x} = 98 - x^{\log 7} \) is approximately:
\[ x \approx 229.08 \]