The value of $e^{{\log }_{e}x+{\log }_{\sqrt{e}}x+{\log }_{\sqrt[3]{e}}x+\dots +{\log }_{10}{\sqrt{e}}^x}$ is

We are tasked to evaluate the expression:

e^log_ex + log_ex + log_√ex + log_√3x + ... + log_√10x    

Step 1: Rewrite the logarithms using the loge value

Since the expression contains logarithms with different bases, we convert all logarithms to the base e. Using the property:

log_ab = log_eb / log_ea    

we can express each term:

• For log_√ex:

  log_√ex = log_ex / log_e√e = log_ex / (1/2) = 2log_ex

• For log_√3x:

  log_√3x = log_ex / log_e√3 = log_ex / (1/3) = 3log_ex

• For log_√4x:

  log_√4x = log_ex / log_e√4 = log_ex / (1/4) = 4log_ex

Continuing in this manner, we rewrite all terms up to log_√10x.

Step 2: Combine all terms

Substituting the rewritten terms into the expression:

e^log_ex + 2log_ex + 3log_ex + 4log_ex + ... + 10log_ex    

Step 3: Factor out loge value

Factoring out log_ex gives:

e^{(1 + 2 + 3 + ... + 10) log_ex}    

Step 4: Calculate the sum of the series

The sum of the first 10 natural numbers is calculated using the formula:

Sum = n(n + 1) / 2        Sum = 10(10 + 1) / 2 = 10 × 11 / 2 = 55    

Step 5: Substitute the sum back

Now the expression becomes:

e^{55 log_ex}    

Step 6: Simplify using logarithmic and exponential properties

We apply the property:

e^log_ea = a    

This simplifies the expression to:

x⁵⁵    

Final Answer

The value of the given expression, after considering all loge values, is: x⁵⁵