Question

given the function below, fill in the table of values, use the table of values to graph the function, and then identify the functions domain and range. $y = \log_{4}x + 4$

Explanation:

Step1: Recall log - rule

Use the formula $\log_a b = c$ means $a^c=b$. For $x = \frac{1}{16}$, we have $\log_4\frac{1}{16}=y$, then $4^y=\frac{1}{16}=4^{- 2}$, so $\log_4\frac{1}{16}=-2$.

Step2: Calculate $y$ - value

Substitute $\log_4\frac{1}{16}=-2$ into $y = \log_4x + 4$. Then $y=-2 + 4=2$.

Step3: For $x=\frac{1}{4}$

We know that if $\log_4\frac{1}{4}=y$, then $4^y=\frac{1}{4}=4^{-1}$, so $\log_4\frac{1}{4}=-1$. And $y=\log_4\frac{1}{4}+4=-1 + 4=3$.

Step4: For $x = 1$

If $\log_41=y$, then $4^y = 1=4^0$, so $\log_41=0$. And $y=\log_41+4=0 + 4=4$.

Step5: For $x = 4$

If $\log_44=y$, then $4^y = 4=4^1$, so $\log_44=1$. And $y=\log_44+4=1 + 4=5$.

Step6: For $x = 10$

$\log_410=\frac{\ln10}{\ln4}\approx\frac{2.3026}{1.3863}\approx1.66$. And $y=\log_410+4\approx1.66+4 = 5.66$.

$x$	$\log_4x$	$y=\log_4x + 4$
$\frac{1}{4}$	$-1$	$3$
$1$	$0$	$4$
$4$	$1$	$5$
$10$	$\approx1.66$	$\approx5.66$

The domain of the function $y=\log_4x + 4$ is $(0,\infty)$ since the argument of the logarithm function $\log_4x$ must be positive. The range of the function is $(-\infty,\infty)$ because the range of the basic logarithmic function $y = \log_4x$ is $(-\infty,\infty)$ and adding a constant $4$ does not change the range.

Answer:

$x$	$\log_4x$	$y=\log_4x + 4$
$\frac{1}{4}$	$-1$	$3$
$1$	$0$	$4$
$4$	$1$	$5$
$10$	$\approx1.66$	$\approx5.66$

Domain: $(0,\infty)$; Range: $(-\infty,\infty)$