Question

14 mark for review in a semi-log plot, which of the following pairs of functions appear linear as parallel lines? a ( f(x) = 2x ) and ( g(x) = 2x + 3 ) b ( f(x) = x^2 ) and ( g(x) = 3x^2 ) c ( f(x) = 2^x ) and ( g(x) = 3 cdot 2^x ) d ( f(x) = ln(2x) ) and ( g(x) = 3ln(2x) )

Explanation:

Step1: Recall semi-log plot linearity

A function \( y = a \cdot b^x \) becomes linear on a semi-log (log-y vs x) plot: \( \log(y) = \log(a) + x\log(b) \), where slope is \( \log(b) \).

Step2: Check parallelism condition

Parallel lines require equal slopes. For option C: \( \log(f(x)) = x\log(2) \), \( \log(g(x)) = \log(3) + x\log(2) \). Both have slope \( \log(2) \), so they are parallel linear lines.

Step3: Eliminate other options

A: Linear on linear plot, not semi-log specific. B: \( \log(y) = 2\log(x) + \log(3) \) (slope 2 vs 2? No, \( g(x)=3x^2 \) gives \( \log(g)=log3+2logx \), but these are power functions, linear on log-log not semi-log. D: \( \log(f) = \log(\ln2x) \), non-linear on semi-log. Only C fits.

Answer:

C. \( f(x) = 2^x \) and \( g(x) = 3 \cdot 2^x \)