The number of solutions of the equation `2x^(log_(10)x)+3x^(log_(10)(1//x))=5` is

Options

A. `1`
B. `2`
C. `3`
D. none of these

Answer

To solve the equation \(2x^{\log_{10} x} + 3x^{\log_{10}(1/x)} = 5\), we can follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We know that \(\log_{10}(1/x) = -\log_{10}(x)\). Therefore, we can rewrite the equation as:
\[ 2x^{\log_{10} x} + 3x^{-\log_{10} x} = 5 \]
This simplifies to:
\[ 2x^{\log_{10} x} + \frac{3}{x^{\log_{10} x}} = 5 \]

### Step 2: Let \(t = x^{\log_{10} x}\)
Now, we substitute \(t\) for \(x^{\log_{10} x}\):
\[ 2t + \frac{3}{t} = 5 \]

### Step 3: Multiply through by \(t\) to eliminate the fraction
Multiplying both sides by \(t\) gives:
\[ 2t^2 + 3 = 5t \]

### Step 4: Rearrange the equation
Rearranging the equation results in:
\[ 2t^2 - 5t + 3 = 0 \]

### Step 5: Factor the quadratic equation
To factor \(2t^2 - 5t + 3\), we can rewrite it as:
\[ 2t^2 - 3t - 2t + 3 = 0 \]
Grouping gives:
\[ t(2t - 3) - 1(2t - 3) = 0 \]
Factoring out \((2t - 3)\):
\[ (2t - 3)(t - 1) = 0 \]

### Step 6: Solve for \(t\)
Setting each factor to zero gives:
1. \(2t - 3 = 0 \Rightarrow t = \frac{3}{2}\)
2. \(t - 1 = 0 \Rightarrow t = 1\)

### Step 7: Back substitute to find \(x\)
Recall that \(t = x^{\log_{10} x}\):
1. For \(t = \frac{3}{2}\):
\[ x^{\log_{10} x} = \frac{3}{2} \]
Taking logarithm on both sides:
\[ \log_{10} x \cdot \log_{10} x = \log_{10} \frac{3}{2} \]
This leads to:
\[ (\log_{10} x)^2 = \log_{10} \frac{3}{2} \]
Therefore:
\[ \log_{10} x = \pm \sqrt{\log_{10} \frac{3}{2}} \]
This gives two values for \(x\).

2. For \(t = 1\):
\[ x^{\log_{10} x} = 1 \]
This implies:
\[ \log_{10} x = 0 \Rightarrow x = 10^0 = 1 \]

### Step 8: Count the number of solutions
From the above, we have:
- Two solutions from \(t = \frac{3}{2}\)
- One solution from \(t = 1\)

Thus, the total number of solutions is \(2 + 1 = 3\).

### Final Answer
The number of solutions of the equation is **3**.

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