The value of
`e^("log"_(e) x+ "log"_(sqrt(e)) x+ "log"_(root(3)(e)) x + …. + "log"_(root(10)(e))x),` is

Options

A. `x^(10)`
B. e
C. `x^(55)`
D. none of these

Explanation

To solve the expression \( e^{\log_e x + \log_{\sqrt{e}} x + \log_{\sqrt[3]{e}} x + \ldots + \log_{\sqrt[10]{e}} x} \), we will follow these steps:

### Step 1: Rewrite the logarithms with a common base
The expression consists of logarithms with different bases. We can convert all of them to a common base, which is \( e \).

\[ \log_{\sqrt{e}} x = \frac{\log_e x}{\log_e \sqrt{e}} = \frac{\log_e x}{\frac{1}{2}} = 2 \log_e x \]

\[ \log_{\sqrt[3]{e}} x = \frac{\log_e x}{\log_e \sqrt[3]{e}} = \frac{\log_e x}{\frac{1}{3}} = 3 \log_e x \]

\[ \log_{\sqrt[4]{e}} x = \frac{\log_e x}{\log_e \sqrt[4]{e}} = \frac{\log_e x}{\frac{1}{4}} = 4 \log_e x \]

Continuing this way, we can express all logarithms up to \( \log_{\sqrt[10]{e}} x \).

### Step 2: Write the complete expression
Now we can write the complete expression as:

\[ e^{\log_e x + 2 \log_e x + 3 \log_e x + 4 \log_e x + 5 \log_e x + 6 \log_e x + 7 \log_e x + 8 \log_e x + 9 \log_e x + 10 \log_e x} \]

### Step 3: Factor out \( \log_e x \)
Factoring out \( \log_e x \):

\[ e^{(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) \log_e x} \]

### Step 4: Calculate the sum of the series
The sum \( 1 + 2 + 3 + \ldots + 10 \) can be calculated using the formula for the sum of the first \( n \) natural numbers:

\[ \text{Sum} = \frac{n(n + 1)}{2} = \frac{10(10 + 1)}{2} = \frac{10 \times 11}{2} = 55 \]

### Step 5: Substitute back into the expression
Now substituting back into the expression, we have:

\[ e^{55 \log_e x} \]

### Step 6: Simplify using properties of exponents and logarithms
Using the property \( e^{\log_e a} = a \):

\[ e^{55 \log_e x} = x^{55} \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{x^{55}} \]

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