Solution:

To evaluate the integral ∫ log x to the base x dx, we can use a technique called integration by manipulation. This involves manipulating the integrand in a way that allows us to apply a known integration rule.

1. Rewrite the logarithmic function:
The first step is to rewrite the logarithmic function in a more convenient form. We can use the change of base formula to rewrite log x to the base x as log x / log x.

2. Substitute u = log x:
Let's substitute u = log x, which means du = (1 / x) dx. This substitution will make the integral easier to evaluate.

3. Rewrite the integral in terms of u:
Using the substitution u = log x, we can rewrite the integral as ∫ (u / log x) (1 / x) dx.

4. Simplify the integral:
Next, we simplify the integral by canceling out the common factors. The expression becomes ∫ (u / x) dx.

5. Apply the power rule:
Now, we can apply the power rule of integration, which states that ∫ x^n dx = (x^(n+1)) / (n+1) + C, where C is the constant of integration.

In this case, the power rule can be applied to the integral ∫ (u / x) dx, where the exponent of x is -1. Applying the power rule, we have:

∫ (u / x) dx = (x^(-1+1)) / (-1+1) + C = ln|x| + C

6. Substitute back u = log x:
Finally, substitute back u = log x into the expression obtained in step 5. The result is:

∫ log x to the base x dx = ln|log x| + C

This is the final result of the integral.

In conclusion, by manipulating the integrand and applying the power rule of integration, we evaluated the integral ∫ log x to the base x dx to be ln|log x| + C.