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Step-by-step explanation:

To find the root mean square (rms) value of log(x) over the range of x from 1 to e, we need to integrate the square of log(x) over that range and then take the square root of the result.

The integral can be expressed mathematically as follows:

RMS value = sqrt(1 / (e - 1) * ∫[1 to e] (log(x))^2 dx)

To solve this integral, we can use integration by parts.

Let's start by applying the formula for integration by parts:

∫ u _ v dx = u _ ∫ v dx - ∫ (u' * ∫ v dx) dx

For this integral, we can choose u = log(x) and dv = log(x) dx.

Differentiating u, we get du = (1 / x) dx.

Integrating dv, we get v = x * log(x) - x.

Now, applying the integration by parts formula, we have:

∫ (log(x))^2 dx = log(x) _ (x _ log(x) - x) - ∫ (x _ log(x) - x) _ (1 / x) dx

Simplifying, we get:

∫ (log(x))^2 dx = x _ (log(x))^2 - x _ log(x) - ∫ (log(x) - 1) dx

Integrating the remaining term, we have:

∫ (log(x))^2 dx = x _ (log(x))^2 - x _ log(x) - (x - x * log(x))

Now, let's substitute the limits of integration, which are 1 and e:

∫[1 to e] (log(x))^2 dx = e _ (log(e))^2 - e _ log(e) - (e - e _ log(e)) - (1 _ (log(1))^2 - 1 _ log(1) - (1 - 1 _ log(1)))

Since log(e) = 1 and log(1) = 0, the equation simplifies to:

∫[1 to e] (log(x))^2 dx = e - e - (e - e) - (0 - 0 - (0 - 0))

Simplifying further, we get:

∫[1 to e] (log(x))^2 dx = 0

Finally, we can calculate the rms value:

RMS value = sqrt(1 / (e - 1) * ∫[1 to e] (log(x))^2 dx)

= sqrt(1 / (e - 1) * 0)

= sqrt(0)

= 0

Therefore, the rms value of log(x) over the range of x from 1 to e is 0.