Describe the transformations of the following graph. Justify your response algebraically.

Question

Describe the transformations that take the graph of  to the graph of . Justify your response algebraically

Answer 1

To solve this problem, you need to know the power rule of logarithms, which states that: log_b(M^k) = k log_b(M)

Use this to rewrite g(x) as the following:

g(x) = log₄(x³) - log₄(8) = 3 log₄(x) - log₄(8)

The tricky red herring here is that using the quotient rule, which allows us to move from the difference of logarithms to a logarithm of a quotient, is NOT helpful for this problem. This rewriting allows us to "find" the original equation of f(x) in the new expression for g(x), which makes it easier to think about the transformation of the graph.

So now, we're ready to transform the graph of f(x) to the graph of g(x)

Step 1 - Original equation of f(x): y = log₄(x)

Step 2 - Stretch the graph by a factor of 3: y = 3log₄(x)

Step 3 - Move the graph down (vertical translation) by log₄(8): y = 3log₄(x) - log₄(8)

Step 4: Recognize that this is the same as the equation for g(x): y = 3 log₄(x) - log₄(8) = log₄(x³) - log₄(8)