Question

The graph of the function f(x)=log4(x) is stretched vertically by a factor of 2, reflected over the x-axis, reflected over the y-axis, and shifted down by 1 unit.Find the equation of the function g(x) described above.

Answer 1

$f(x)=\log_4x$

Transformations on a lograrithmic function:

$\begin{gathered} Reflections: \\ over\text{ x-axis: -}\log_bx \\ over\text{ y-axis:}\log_b(-x) \end{gathered}$$\begin{gathered} \text{Vertical stretching: }a\log_bx\text{ \lparen a is the factor of streatching\rparen} \\ \\ \end{gathered}$$Shift\text{ down k units:}\log_b(x)-k$

Then, the given function after the given transformations is:

$g(x)=-2\log_4(-x)-1$