Question

function
logarithmic function:
$f(x)=log x$
left end behavior (use limit notation):
horizontal asymptote(s): (write in the form of y = __ )
sketch of graph
right end behavior (use limit notation):
vertical asymptote(s). (write in the form of x = __ )
write using interval notation
intervals of increasing
intervals of decreasing
write using interval notation.
domain:
write using interval notation
range:

Explanation:

Step1: Analyze domain

The logarithm function $y = \log x$ is defined for $x>0$. So the domain is $(0,\infty)$.

Step2: Analyze range

The range of the common - logarithm function is all real numbers, i.e., $(-\infty,\infty)$.

Step3: Analyze end - behaviors

As $x\to0^{+}$, $\lim_{x\to0^{+}}\log x=-\infty$ (left - end behavior). As $x\to\infty$, $\lim_{x\to\infty}\log x=\infty$ (right - end behavior).

Step4: Analyze asymptotes

The vertical asymptote is $x = 0$ since the function is not defined for $x\leq0$ and the limit as $x\to0^{+}$ is $-\infty$. There is no horizontal asymptote.

Step5: Analyze increasing and decreasing intervals

The derivative of $y=\log x$ is $y'=\frac{1}{x\ln10}$. Since $y'>0$ for $x > 0$, the function is increasing on the interval $(0,\infty)$ and there is no interval of decreasing.

Answer:

• Intervals of increasing: $(0,\infty)$
• Intervals of decreasing: None
• Left End Behavior: $\lim_{x\to0^{+}}\log x=-\infty$
• Right End Behavior: $\lim_{x\to\infty}\log x=\infty$
• Domain: $(0,\infty)$
• Horizontal Asymptote(s): None
• Vertical Asymptote(s): $x = 0$
• Range: $(-\infty,\infty)$