Graphing Logarithmic Functions - Algebra 2

Card 0 of 28

Question

Give the intercept of the graph of the function

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

to two decimal places.

Answer

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

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Question

Give the -intercept of the graph of the function

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

to two decimal places.

Answer

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

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Question

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

Answer

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

.

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Question

Which is true about the graph of

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

Answer

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.

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Question

Which of the following is true about the graph of

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

Answer

$\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 $\small y=x$

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Question

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

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

Answer

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 .

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Question

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

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

Answer

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.

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Question

Give the intercept of the graph of the function

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

to two decimal places.

Answer

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

Compare your answer with the correct one above

Question

Give the -intercept of the graph of the function

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

to two decimal places.

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

.

Compare your answer with the correct one above

Question

Which is true about the graph of

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

Answer

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.

Compare your answer with the correct one above

Question

Which of the following is true about the graph of

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

Answer

$\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 $\small y=x$

Compare your answer with the correct one above

Question

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

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

Answer

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 .

Compare your answer with the correct one above

Question

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

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

Answer

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.

Compare your answer with the correct one above

Question

Give the intercept of the graph of the function

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

to two decimal places.

Answer

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

Compare your answer with the correct one above

Question

Give the -intercept of the graph of the function

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

to two decimal places.

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

.

Compare your answer with the correct one above

Question

Which is true about the graph of

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

Answer

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.

Compare your answer with the correct one above

Question

Which of the following is true about the graph of

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

Answer

$\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 $\small y=x$

Compare your answer with the correct one above

Question

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

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

Answer

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 .

Compare your answer with the correct one above

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