To determine the correct transformation of the function \( f(x) = \log_2 x \) based on the given conditions, we must apply the transformations step-by-step.
-
Shift 7 units to the right: This shifts the graph horizontally. The transformation for shifting \( f(x) \) to the right by 7 units is: \[ f(x) \to f(x - 7) = \log_2(x - 7). \]
-
Shrink vertically by a factor of 13: This transformation multiplies the function by \( \frac{1}{13} \): \[ f(x - 7) \to \frac{1}{13} f(x - 7) = \frac{1}{13} \log_2(x - 7). \]
-
Shift 5 units up: Adding 5 to the function gives: \[ \frac{1}{13} f(x - 7) + 5 = \frac{1}{13} \log_2(x - 7) + 5. \]
Thus, the transformed function after all the transformations is: \[ g(x) = \frac{1}{13} \log_2(x - 7) + 5. \]
None of the given options match exactly, as we derived a vertical shrinking by a factor of \(\frac{1}{13}\). We can examine the options, keeping in mind we to multiply the output of the log function by \(\frac{1}{13}\) instead of \(13\).
Let's analyze each option:
-
Option #1: \( g(x) = 3\log_2(x+5) - 7 \) does not represent the correct transformations because it has the wrong transformations and shifts.
-
Option #2: \( g(x) = 13\log_2(x+7) + 5 \) shifts in the wrong direction, and uses the wrong factor.
-
Option #3: \( g(x) = 13\log_2(x−7) + 5 \) applies an incorrect vertical stretch (should be a shrink, not stretch).
Therefore, none of the options provided correctly represent the transformations applied to \( f(x) = \log_2 x \). The key error is that we are supposed to shrink by a factor of 13 (leading to \( \frac{1}{13} \)) while all given options incorrectly would expand the log function vertically instead of shrinking it.
Hence, based on the transformations described, the correct function would be: \[ g(x) = \frac{1}{13} \log_2(x - 7) + 5. \] Since this function is not among the provided options, none is correct.