Question - Logarithmic Function Transformation Analysis
Solution:
The logarithmic function $$ f(x) = \ln(x) $$ is transformed into $$ g(x) = \ln(x - 2) + 5 $$. To determine which of the provided statements is true, we need to analyze the transformation in two parts: the horizontal shift caused by the change inside the logarithm and the vertical shift caused by the addition outside the logarithm.The $$ \ln(x) $$ function is shifted horizontally by the presence of $$-2$$ inside the function, which means the function is shifted to the right by 2 units. That's because the logarithm’s argument $$ x $$ is replaced with $$ x - 2 $$, indicating a rightward shift of 2 units.On the other hand, adding $$ +5 $$ to the function after the logarithm $$ \ln(x - 2) $$ indicates a vertical shift. Specifically, it means the graph of the function is shifted upwards by 5 units.Given these transformations:A. Incorrect: $$ f(x) $$ is not translated up 5 units; it's the function $$ g(x) $$ that has been translated up by 5 units compared to $$ f(x) $$.B. Incorrect: $$ f(x) $$ is not translated down 2 units.C. Incorrect: $$ f(x) $$ is not translated left 2 units; it's moving to the right.D. Correct: $$ f(x) $$ is translated right 5 units, which exactly describes the horizontal shift.Therefore, the correct answer is D: $$ f(x) $$ is translated right 5 units.
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