Find 'x' satisfying the equation
`4^(log_(10) x + 1) - 6^(log_(10)x) - 2.3 ^(log_(10)x^(2) + 2) = 0 .`

Answer

To solve the equation

\[ 4^{(\log_{10} x + 1)} - 6^{(\log_{10} x)} - 2 \cdot 3^{(\log_{10} x^2 + 2)} = 0, \]

we will follow these steps:

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ 4^{\log_{10} x} \cdot 4^1 - 6^{\log_{10} x} - 2 \cdot 3^{(\log_{10} x^2 + 2)} = 0. \]

This simplifies to:

\[ 4 \cdot 4^{\log_{10} x} - 6^{\log_{10} x} - 2 \cdot 3^{(2 \log_{10} x + 2)} = 0. \]

### Step 2: Simplify the terms
Using the logarithmic identity \(a^{\log_b c} = c^{\log_b a}\):

- \(4^{\log_{10} x} = x^{\log_{10} 4}\)
- \(6^{\log_{10} x} = x^{\log_{10} 6}\)
- \(3^{(2 \log_{10} x + 2)} = 3^{2 \log_{10} x} \cdot 3^2 = 9 \cdot x^{\log_{10} 9}\)

Substituting these back into the equation gives:

\[ 4 \cdot x^{\log_{10} 4} - x^{\log_{10} 6} - 18 \cdot x^{\log_{10} 3} = 0. \]

### Step 3: Factor out common terms
Let \(y = x^{\log_{10} x}\). The equation becomes:

\[ 4y^{\log_{10} 4} - y^{\log_{10} 6} - 18y^{\log_{10} 3} = 0. \]

### Step 4: Substitute \(t = \left(\frac{2}{3}\right)^{\log_{10} x}\)
Let \(t = \left(\frac{2}{3}\right)^{\log_{10} x}\). Then we can rewrite the equation as:

\[ 4t^2 - t - 18 = 0. \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 4\), \(b = -1\), and \(c = -18\):

\[ t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 4 \cdot (-18)}}{2 \cdot 4} \]

Calculating the discriminant:

\[ 1 + 288 = 289 \]

So:

\[ t = \frac{1 \pm 17}{8}. \]

This gives us two solutions:

1. \(t = \frac{18}{8} = \frac{9}{4}\)
2. \(t = \frac{-16}{8} = -2\)

### Step 6: Discard the negative solution
Since \(t\) represents an exponential function, it must be positive. Therefore, we discard \(t = -2\).

### Step 7: Solve for \(x\)
Now we have:

\[ \left(\frac{2}{3}\right)^{\log_{10} x} = \frac{9}{4}. \]

Taking logarithm on both sides:

\[ \log_{10} x = \log_{10} \left(\frac{9}{4}\right) \cdot \frac{1}{\log_{10} \left(\frac{2}{3}\right)}. \]

### Step 8: Calculate \(x\)
Using properties of logarithms:

\[ x = 10^{\log_{10} \left(\frac{9}{4}\right) \cdot \frac{1}{\log_{10} \left(\frac{2}{3}\right)}}. \]

This leads to:

\[ x = \frac{9}{4} \text{ (as we can simplify further)}. \]

### Final Result
Thus, the solution for \(x\) is:

\[ x = \frac{1}{100} = 0.01. \]