Question

when graphed on a semi-log plot, which of the following best describes the graphs of f(x) = (4)^x and g(x) = 6(4)^x? both f and g appear linear and have the same rate of change so their graphs are parallel lines. both f and g appear linear and have the same rate of change so their graphs are perpendicular lines. both f and g appear linear and intersecting because g will have a larger rate of change than f. both f and g appear quadratic and have the same rate of change so their graphs are parallel lines.

Explanation:

Step1: Take log of f(x)

$\log(f(x)) = \log(4^x) = x\log(4)$

Step2: Take log of g(x)

$\log(g(x)) = \log(6(4^x)) = \log(6) + x\log(4)$

Step3: Analyze linear forms

Both are linear in $x$ with slope $\log(4)$ (same rate of change). Linear functions with equal slope are parallel.

Answer:

Both f and g appear linear and have the same rate of change so their graphs are parallel lines.