Question

1. an electronic designer wants to determine the decibel rating of his companys new room fan using a logarithmic decibel function.

$p(x)=15\log(x)$, where

$p(x)=db=$ the intensity (loudness) of a sound measured in decibels, and

$x=$ the number of times a given sound is to the threshold sound.

a) using the following graph of $p(x)=15\log(x)$, what is the right - end behavior and left - end behavior for this logarithmic function written with infinity notation?

1. right - end behavior:

1. left - end behavior:

Explanation:

Step1: Analyze right - end behavior

For the function \(P(x)=15\log(x)\), the domain of the logarithmic function \(y = \log(x)\) is \(x>0\). As \(x\) approaches positive infinity (\(x
ightarrow+\infty\)), we know that the logarithmic function \(y = \log(x)\) is an increasing function (since the base of the common logarithm is greater than 1). When we multiply the logarithmic function by 15 (a positive constant), the behavior of the function \(P(x)\) as \(x
ightarrow+\infty\) will be the same as the behavior of \(\log(x)\) as \(x
ightarrow+\infty\) in terms of the direction of increase. So, as \(x
ightarrow+\infty\), \(\log(x)
ightarrow+\infty\), and then \(P(x)=15\log(x)
ightarrow+\infty\). In limit notation, \(\lim_{x
ightarrow+\infty}P(x)=\lim_{x
ightarrow+\infty}15\log(x)=+\infty\).

Step2: Analyze left - end behavior

The domain of \(y = \log(x)\) is \(x > 0\). As \(x\) approaches 0 from the right side (\(x
ightarrow0^{+}\)), we know that \(\log(x)\) approaches \(-\infty\) (because when \(x\) gets closer and closer to 0 while being positive, the value of \(\log(x)\) becomes more and more negative). Then, for the function \(P(x)=15\log(x)\), when \(x
ightarrow0^{+}\), \(15\log(x)\) will also approach \(-\infty\) (since we are multiplying a quantity that approaches \(-\infty\) by a positive constant 15). In limit notation, \(\lim_{x
ightarrow0^{+}}P(x)=\lim_{x
ightarrow0^{+}}15\log(x)=-\infty\). And since the domain of the function does not include \(x\leq0\), we only consider the left - end behavior as \(x\) approaches 0 from the right.

Answer:

1. right - end behavior: \(\boldsymbol{\lim_{x

ightarrow+\infty}15\log(x)=+\infty}\) (or as \(x
ightarrow+\infty\), \(P(x)
ightarrow+\infty\))

1. left - end behavior: \(\boldsymbol{\lim_{x

ightarrow0^{+}}15\log(x)=-\infty}\) (or as \(x
ightarrow0^{+}\), \(P(x)
ightarrow-\infty\))