IB MATH AA HL, Paper 3, May, 2023, TZ2, Solved Past Paper

Examine the cases where a = 2 and a = 10. On a single coordinate system, draw the graphs for the following equations:

y = log₂(x)

y = log₁₀(x)

y = x

Ensure each curve is clearly identified with its respective formula. Also, determine and specify any non-zero points where these graphs intersect the x-axis.

For parts (b) and (c), consider the scenario where (a = e). Recall that (ln x) is the natural logarithm, or (log_e x).

Utilize calculus to determine the minimum value of the function (x - ln x). Ensure your solution confirms that this is indeed the minimum value.

Let's consider the inequality (x > ln(x)).

In this scenario, we are asked to explore under what conditions the value of (x), a real number, surpasses that of its natural logarithm, (ln(x)). This involves examining the relationship between a linear growth model and a logarithmic model to determine where the linear value exceeds the logarithmic one. This comparison is central to understanding the behaviors of these functions in relation to each other.