Given the functions 𝑓(𝑥) = 𝑙𝑜𝑔_3 (3𝑥) and 𝑔(𝑥) = 𝑙𝑜𝑔_3 (𝑥) + 1
a. Describe the transformations applied to each function.
b. How do the graphs of the two functions compare? Explain your answer by referring to logarithmic
laws and properties.
log3(3x) = log3(3) + log3(x) = 1 + log3(x)
f(x) is identical to g(x)
g(x) is f(x)
dilated in x by 3
shifted up 1
The two transformations cancel each other out
the graphs of all exponential functions look the same
dilating and shifting are complementary operations.
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.