Question

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)=3cdot2^{x} )
d ( f(x)=ln(2x) ) and ( g(x)=3ln(2x) )

Explanation:

Step1: Recall the property of semi - log plot

In a semi - log plot (where \(y\) is on the log scale and \(x\) is on the linear scale), if \(y = a\cdot b^{x}\), then \(\ln y=\ln a + x\ln b\). The equation \(\ln y = mx + c\) (where \(m = \ln b\) and \(c=\ln a\)) represents a straight line. For two lines \(y_1=a_1\cdot b^{x}\) and \(y_2=a_2\cdot b^{x}\) (same base \(b\) of the exponential function), \(\ln y_1=\ln a_1+x\ln b\) and \(\ln y_2=\ln a_2 + x\ln b\). The slopes (\(m=\ln b\)) are equal.

Step2: Analyze each option

• Option A:

\(f(x)=2x\) and \(g(x)=2x + 3\) are linear functions. In a semi - log plot (not a standard Cartesian plot), \(y = 2x\) and \(y=2x + 3\) do not follow the form \(y=a\cdot b^{x}\). The transformation for a semi - log plot (\(y\) on log - scale) will not make them parallel lines in the context of semi - log plot properties for exponential - like functions.

• Option B:

\(f(x)=x^{2}\) and \(g(x)=3x^{2}\). If \(y = x^{2}\), taking \(\ln y=2\ln x\). If \(y = 3x^{2}\), \(\ln y=\ln 3+2\ln x\). But in a semi - log plot (\(y\) on log - scale, \(x\) on linear - scale), the relationship is not of the form \(y=a\cdot b^{x}\) (it's a power function, not an exponential function).

• Option C:

For \(f(x)=2^{x}\), \(\ln f(x)=x\ln 2\). For \(g(x)=3\cdot2^{x}\), \(\ln g(x)=\ln 3+x\ln 2\). The equations \(\ln f(x)\) and \(\ln g(x)\) are of the form \(y = mx + c\) (where \(m=\ln 2\) for both). Since the slopes (\(m\)) are equal (and the \(y\) - intercepts (\(c\)) are different: \(c = 0\) for \(y=\ln f(x)\) and \(c=\ln 3\) for \(y=\ln g(x)\)), they are parallel lines in a semi - log plot (\(y\) on log - scale).

• Option D:

\(f(x)=\ln(2x)=\ln 2+\ln x\) and \(g(x)=3\ln(2x)=3\ln 2 + 3\ln x\). In a semi - log plot (\(y\) on log - scale, \(x\) on linear - scale), they do not follow the \(y=a\cdot b^{x}\) (and subsequent \(\ln y=\ln a+x\ln b\)) form.

Answer:

C. \(f(x)=2^{x}\) and \(g(x)=3\cdot2^{x}\)