To describe the transformations applied to each function, let's start by examining the original functions before the transformations:
The original function f(x) = log₃(3x) represents a logarithmic function with base 3.
The original function g(x) = log₃(x) + 1 also represents a logarithmic function with base 3, but with an additional constant 1 added to it.
To analyze the transformations, we can focus on the differences between the original and transformed functions:
a. Transformations applied to f(x) = log₃(3x):
- The function f(x) = log₃(3x) has undergone a vertical stretch transformation.
- The stretch is caused by multiplying the argument of the logarithm, 3x, by 3. This change causes the graph of the function to expand vertically by a factor of 3.
b. Transformations applied to g(x) = log₃(x) + 1:
- The function g(x) = log₃(x) + 1 has undergone a vertical shift transformation.
- The shift is caused by adding the constant 1 to the entire function. This change shifts the graph of the function upward by 1 unit.
Now, let's compare the graphs of f(x) and g(x) and explain how they relate to each other using logarithmic laws and properties:
The graph of f(x) = log₃(3x) is a shifted and compressed version of the graph of g(x) = log₃(x). The original graph, g(x), represents the logarithm of x to the base 3. By multiplying 3x by 3, we compress the graph of f(x) horizontally. However, since we are applying a logarithmic function, the logarithmic law states that logₐ(b * c) = logₐ(b) + logₐ(c). In this case, we have log₃(3x) = log₃(3) + log₃(x). The term log₃(3) can be simplified to 1, so f(x) simplifies to log₃(x) + 1, which is equivalent to g(x) = log₃(x) + 1. Therefore, the transformation applied to f(x) is not only a compression but also a vertical shift due to the logarithmic law.
In summary, the graph of f(x) = log₃(3x) is a vertically stretched version of g(x) = log₃(x), with the addition of a vertical shift of 1 unit. The transformations applied to each function are a vertical stretch and a vertical shift, respectively.