Solve the following equations.
(vii) `4^(log_10x+1)-6^(log_10x)-2.3^(log_10x^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 can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4^{(\log_{10} x + 1)} - 6^{\log_{10} x} - 2 \cdot 3^{(\log_{10} x^2 + 2)} = 0 \]
This can be rewritten as:
\[ 4^{\log_{10} x} \cdot 4 - 6^{\log_{10} x} - 2 \cdot 3^{(\log_{10} x^2)} \cdot 3^2 = 0 \]

### Step 2: Substitute \(t\)
Let \(t = \log_{10} x\). Then we can rewrite the equation as:
\[ 4 \cdot 4^t - 6^t - 2 \cdot 9 \cdot 3^{2t} = 0 \]
This simplifies to:
\[ 4^{t+1} - 6^t - 18 \cdot 3^{2t} = 0 \]

### Step 3: Express in terms of powers
We know that:
\[ 4^t = (2^2)^t = 2^{2t}, \quad 6^t = (2 \cdot 3)^t = 2^t \cdot 3^t, \quad 3^{2t} = (3^t)^2 \]
Substituting these into the equation gives:
\[ 4 \cdot 2^{2t} - 2^t \cdot 3^t - 18 \cdot (3^t)^2 = 0 \]

### Step 4: Divide by \(3^{2t}\)
Now, divide the entire equation by \(3^{2t}\):
\[ \frac{4 \cdot 2^{2t}}{3^{2t}} - \frac{2^t \cdot 3^t}{3^{2t}} - 18 = 0 \]
This simplifies to:
\[ 4 \cdot \left(\frac{2}{3}\right)^{2t} - 2 \cdot \left(\frac{2}{3}\right)^{t} - 18 = 0 \]

### Step 5: Let \(u = \left(\frac{2}{3}\right)^{t}\)
Let \(u = \left(\frac{2}{3}\right)^{t}\). Then, we can rewrite the equation as:
\[ 4u^2 - 2u - 18 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 4\), \(b = -2\), and \(c = -18\).
\[ u = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 4 \cdot (-18)}}{2 \cdot 4} \]
\[ u = \frac{2 \pm \sqrt{4 + 288}}{8} \]
\[ u = \frac{2 \pm \sqrt{292}}{8} \]
\[ u = \frac{2 \pm 17.09}{8} \]
Calculating the two possible values for \(u\):
1. \(u_1 = \frac{19.09}{8} \approx 2.38625\)
2. \(u_2 = \frac{-15.09}{8} \approx -1.88625\) (not valid since \(u\) must be positive)

### Step 7: Solve for \(t\)
Now, we have:
\[ \left(\frac{2}{3}\right)^{t} \approx 2.38625 \]
Taking logarithm on both sides:
\[ t \log\left(\frac{2}{3}\right) = \log(2.38625) \]
Since \(\log\left(\frac{2}{3}\right)\) is negative, we can write:
\[ t = \frac{\log(2.38625)}{\log\left(\frac{2}{3}\right)} \]

### Step 8: Substitute back to find \(x\)
Recall that \(t = \log_{10} x\):
\[ \log_{10} x = t \]
Thus,
\[ x = 10^{t} \]

### Final Answer
Calculating the value of \(x\) gives:
\[ x = 10^{\frac{\log(2.38625)}{\log\left(\frac{2}{3}\right)}} \]