Solve the following equations.
(i) `x^(1+log_10x)=10x`

Answer

To solve the equation \( x^{1 + \log_{10} x} = 10x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^{1 + \log_{10} x} = 10x \]

### Step 2: Simplify the left-hand side
Using the properties of exponents, we can rewrite the left-hand side:
\[ x^{1 + \log_{10} x} = x^1 \cdot x^{\log_{10} x} = x \cdot 10 \]
This is because \( x^{\log_{10} x} = 10 \) (from the definition of logarithms).

### Step 3: Set the equation
Now we can set the equation:
\[ x \cdot 10 = 10x \]

### Step 4: Simplify the equation
We can simplify this equation:
\[ 10x = 10x \]
This is always true, which means the equation holds for all values of \( x \) except where it is undefined.

### Step 5: Determine the domain
Since we are dealing with logarithms, we need to ensure that \( x > 0 \) (as logarithms are only defined for positive numbers).

### Conclusion
Thus, the solution to the equation \( x^{1 + \log_{10} x} = 10x \) is:
\[ \text{All } x > 0 \]