Question

1. extend a concept

consider the functions
$f(x) = \log_{4}(2x + 4)$ and $g(x) = \ln(x) + \frac{1}{2}$.
use your calculator to answer the following questions.
a) find the point of intersection.
(3.3550, 1.7194)
b) for what values of $x$ is $f(x) \leq 0$?
$f(x) = -1.5 \geq$
$g(x) = 0.6065 \geq$

Explanation:

Step1: Solve part (a) intersection

Set $f(x)=g(x)$: $\log_4(2x+4) = \ln(x)+\frac{1}{2}$. Use calculator to solve for $x$, then find $y$.

1. Rewrite $\log_4(2x+4)=\frac{\ln(2x+4)}{\ln(4)}$
2. Solve $\frac{\ln(2x+4)}{\ln(4)} - \ln(x) = \frac{1}{2}$ numerically.

Step2: Solve part (b) $f(x)\leq0$

Set $\log_4(2x+4)\leq0$. Convert to exponential form:
$0 < 2x+4 \leq 4^0=1$
Solve inequality:

1. $2x+4 > 0 \implies x > -2$
2. $2x+4 \leq1 \implies 2x \leq -3 \implies x \leq -\frac{3}{2}=-1.5$

Answer:

a) $(3.3550, 1.7104)$
b) $-2 < x \leq -1.5$