(a) Sketch of \( y = \log_2 x \), \( y = \log_{10} x \), and \( y = x \):

Correct shapes: \( y = \log_2 x \) and \( y = \log_{10} x \) are increasing, concave downward with vertical asymptote at \( x = 0 \); \( y = x \) is a straight line with slope 1: (A1)

Label all graphs clearly: (A1)

Non-zero x-intercept: Both logarithmic graphs pass through \( (1, 0) \): (A1)

[3 marks]

(b) Minimum value of \( x – \ln x \):

Differentiate \( f(x) = x – \ln x \):

\[ f'(x) = 1 – \frac{1}{x} \quad (A1) \]

Find critical points: Set \( f'(x) = 0 \):

\[ 1 – \frac{1}{x} = 0 \implies x = 1 \quad (A1) \]

Second derivative test: \( f”(x) = \frac{1}{x^2} \), so \( f”(1) = 1 > 0 \), confirming a local minimum:

\[ f(1) = 1 – \ln 1 = 1 \quad (A1) \]

[3 marks]

(c) Deduce \( x > \ln x \):

From part (b), \( x – \ln x \geq 1 > 0 \), so \( x > \ln x \) for all \( x \in \mathbb{R}^+ \): (A1)

[1 mark]

(d) Number of intersection points:

For \( 0 < a < 1 \): \( y = \log_a x \) is decreasing, intersects \( y = x \) at \( x = 1 \): \( p = 1 \quad (A1) \)

For \( 1 < a < 1.4 \): \( y = \log_a x \) is increasing, intersects \( y = x \) at \( x = 1 \) and another point: \( q = 2 \quad (A1) \)

For \( 1.5 < a < 2 \): \( y = \log_a x \) grows slower, no intersections: \( r = 0 \quad (A1) \)

\[ p = 1, q = 2, r = 0 \quad (A1) \]

[3 marks]

(e) Tangency point \( P \) and value of \( a \):

At tangency, slopes equal: \( \frac{1}{x \ln a} = 1 \implies x = \frac{1}{\ln a} \quad (A1) \)

Intersection: \( \log_a x = x \implies \frac{\ln x}{\ln a} = x \), substitute \( x = \frac{1}{\ln a} \):

\[ \ln x = 1 \implies x = e \quad (A1) \]

Solve for \( a \): \( e = \frac{1}{\ln a} \implies \ln a = \frac{1}{e} \implies a = e^{1/e} \quad (A1) \)

Coordinates: \( y = x = e \), so \( P = (e, e) \quad (A1) \)

[3 marks]

(f) Values of \( a \) for intersection points:

(i) Two intersection points: \( 1 < a < e^{1/e} \quad (A1) \)

(ii) No intersection points: \( a > e^{1/e} \quad (A1) \)

[2 marks]