The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2.

Step 1: Write the integral
The area is calculated as:

A = ∫₁² log x dx

Step 2: Integrate
This can be evaluated as a result of integration by parts. Note that the integration by parts formula is as follows:
∫ u dv = uv – ∫ v du
Here let u = log x and dv = dx, then du = (1/x) dx, v = x
Thus
∫ log x dx = x log x – ∫ x (1/x) dx = x log x – x
Step 3: Calculate the result
Now integrate the integral from x = 1 to x = 2:

A = [x log x – x]₁²

When x = 2,

2 log 2 – 2

When x = 1,

1 log 1 – 1 = 0 – 1 = -1

Hence area is:

A = (2 log 2 – 2) – (-1) = 2 log 2 – 2 + 1 = 2 log 2 – 1

Now substituting the value for log 2 = 0.693.

A ≈ 2(0.693) – 1 = 1.386 – 1 = 0.386

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