Graphing Logarithmic Functions

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Algebra II › Graphing Logarithmic Functions

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Give the -intercept of the graph of the function

$f (x) = \ln (x + 4 ) - 7$

to two decimal places.

$\left ( 0 , -5.61 \right )$

CORRECT

The graph has no -intercept.

$\left (0, 1 092.63 \right )$

$\left ( 0, 20.09 \right )$

$\left (0, 5.75 \right )$

Explanation

Set and solve:

$f(x) = \ln (x + 4 ) - 7$

$f(x) = \ln ( 0 + 4 ) - 7 = \ln 4 - 7 \approx 1.39 - 7 \approx -5.61$

The -intercept is $\left ( 0 , -5.61 \right )$ .

Give the intercept of the graph of the function

$f (x) = \ln (x + 4 ) - 7$

to two decimal places.

$\left (1 092.63, 0 \right )$

CORRECT

$\left ( 20.09, 0 \right )$

$\left ( -5.61,0 \right )$

$\left ( 5.75,0 \right )$

The graph has no -intercept.

Explanation

Set and solve:

$\ln (x + 4 ) - 7 = 0$

$\ln (x + 4 ) =7$

$e ^{\ln (x + 4 ) } =e^{7}$

$x + 4 \approx 1,096.63$

$x \approx 1,092.63$

The -intercept is $\left (1,092.63, 0 \right )$ .

What is/are the asymptote(s) of the graph of the function $f(x) = \ln (x + 6) - 7$ ?

CORRECT

and

Explanation

The graph of the logarithmic function

$\ln (x - h) + k$

has as its only asymptote the vertical line

Here, since , the only asymptote is the line

Which is true about the graph of

$\small f(x)=2 \log(x)$ ?

All of the answers are correct

CORRECT

The domain of the function is greater than zero

The range of the function is infinite in both directions positive and negative.

When $\small x>1$ , $\small y$ is twice the size as in the equation $\small f(x)=log(x)$

None of the answers are correct

Explanation

There is no real number $\small a$ for which $\small \small \small 10^{a}\leq0$

Therefore in the equation $\small log(x)$ , $\small x$ cannot be $\small \leq0$

However, $\small y$ can be infinitely large or negative.

Finally, when $\small x>0$ $\small 2\log (x)=2\times \log (x)$ or twice as large.

Which of the following is true about the graph of

$\small \small y= \log_{2}(x)$

The graph is the mirror image of $\small \small y=2^{x}$ flipped over the line $\small \small y=x$

CORRECT

The domain is infinite in both directions.

The range must be greater than zero.

It is an even function.

It is an odd function.

Explanation

$\small y=\small \small \log_{2} (x)$ is the inverse of $\small y=2^{x}$ and therefore the graph is simply the mirror image flipped over the line $\small y=x$

Give the equation of the horizontal asymptote of the graph of the equation

$f(x) =3 \ln (x+ 4)+ 2$ .

The graph of does not have a horizontal asymptote.

CORRECT

Explanation

Let $g(x) = \ln x$

In terms of ,

This is the graph of shifted left 4 units, stretched vertically by a factor of 3, then shifted up 2 units.

The graph of does not have a horizontal asymptote; therefore, a transformation of this graph, such as that of , does not have a horizontal asymptote either.

Find the equation of the vertical asymptote of the graph of the equation

$f(x) =3 \ln (x+ 4)+ 2$ .

CORRECT

Explanation

Let $g(x) = \ln x$ . In terms of ,

The graph of has as its vertical asymptote the line of the equation . The graph of is the result of three transformations on the graph of - a left shift of 4 units , a vertical stretch ( ), and an upward shift of 2 units ( ). Of the three transformations, only the left shift affects the position of the vertical asymptote - the asymptote of also shifts left 4 units, to .