Question

Let f (x) = log2(x) + 2 and g(x) = log2(x3) – 4.Part A: If h(x) = f (x) + g(x), solve for h(x) in simplest form. (4 points)Part B: Determine the solution to the system of nonlinear equations. (6 points)

Answer 1

Given the functions f(x) and g(x) defined as:

$\begin{gathered} f(x)=\log _2(x)+2 \\ g(x)=\log _2(x^3)-4 \end{gathered}$

A)

We define the function h(x) as:

$h(x)=f(x)+g(x)$

Using the definitions of f(x) and g(x):

$\begin{gathered} h(x)=\log _2(x)+2+\log _2(x^3)-4=\log _2(x\cdot x^3)-2 \\ \Rightarrow h(x)=4\log _2(x)-2 \end{gathered}$

Where we used the following properties of logarithms:

$\begin{gathered} \log _2(a^b)=b\cdot\log _2(a) \\ \log _2(a)+\log _2(b)=\log _2(a\cdot b) \end{gathered}$