Question

when graphed on a semi-log plot, which of the following best describes the graphs of \\(f(x) = (5)^x\\) and \\(g(x) = 2(5)^x\\)?

both \\(f\\) and \\(g\\) are linear and intersecting because \\(g\\) will have a larger rate of change than \\(f\\).
both \\(f\\) and \\(g\\) are linear and have the same rate of change so their graphs are perpendicular lines.
both \\(f\\) and \\(g\\) are linear and have the same rate of change so their graphs are parallel lines.
both \\(f\\) and \\(g\\) are quadratic and have the same rate of change so their graphs are parallel lines.

Explanation:

Apply logarithmic transformation

Using the Exponential Equations and Logarithmic Models knowledge points

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Compare linear properties

Using the Logarithmic Models knowledge point

$$
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Determine geometric relationship

Since both transformed functions are linear equations with identical slopes but different vertical intercepts, their graphs on a semi-log plot are parallel lines.

Answer:

• (A) Both \(f\) and \(g\) are linear and intersecting because \(g\) will have a larger rate of change than \(f\).
• (B) Both \(f\) and \(g\) are linear and have the same rate of change so their graphs are perpendicular lines.
• (C) Both \(f\) and \(g\) are linear and have the same rate of change so their graphs are parallel lines. (Correct answer)
• (D) Both \(f\) and \(g\) are quadratic and have the same rate of change so their graphs are parallel lines.