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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To determine the area bounded by the curve y = 2ˣ, the x-axis, and the ordinates x = 0 and x = 4, we are required to compute the definite integral of 2ˣ from x = 0 to x = 4.

Step 1: Write down the integral
The area is given by:

A = ∫₀⁴ 2ˣ dx

Step 2: Evaluate the integral
The integral of 2ˣ is:

∫ 2ˣ dx = (2ˣ) / ln 2

Now, calculate the area under the curve from x = 0 to x = 4:

A = [(2ˣ) / ln 2]₀⁴

At x = 4:

(2⁴) / ln 2 = 16 / ln 2

At x = 0:

(2⁰) / ln 2 = 1 / ln 2

Therefore, the area:

A = (16 / ln 2) – (1 / ln 2) = 15 / ln 2

Step 3: Final result
So the region is:

A = 15 / ln 2 square units

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