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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #76 : Solving Logarithms

Solve log_5(3x-8)=2 .

Possible Answers:

Correct answer:

Explanation:

We start by rewriting the equation in exponential form:

3x-8=5^2

Now we can simplify:

3x-8=25

3x=33

x=11

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Example Question #211 : Logarithms

Solve log (2x) = log (24) .

Possible Answers:

Correct answer:

Explanation:

We can either do this the long and proper way, or the simple and easy way.

The long way:

First, we move both logs to the same side of the equation:

log (2x)- log (24) =0

Now we can combine the logs (reminder, when you subtract logs, you divide the terms inside of them):

$log(\frac{(2x)}{24})=0$

Let's rewrite the equation in exponential form:

$\frac{2x}{24}=10^0$

Anything raised to the power equals , so we can simplify and solve from here:

$\frac{2x}{24}=1$

2x=24

x=12

The short way:

First, we cancel the log terms (because the base is the same, and all we have are the log terms):

2x=24

Then we divide by :

x=12

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Example Question #3121 : Algebra Ii

$\log_{8}64$ = _______

Possible Answers:

Correct answer:

Explanation:

$\log_{b}a$ is equal to $b^{x}=a$ ,

so in this case it is

$8^{x}=64$ ,

and

x=2

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Example Question #1 : Graphing Logarithmic Functions

Give the -intercept of the graph of the function

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

to two decimal places.

Possible Answers:

$\left ( 0, 20.09 \right )$

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

$\left (0, 5.75 \right )$

The graph has no -intercept.

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

Correct answer:

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

Explanation:

Set x = 0 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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Example Question #1 : Graphing Logarithmic Functions

Give the intercept of the graph of the function

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

to two decimal places.

Possible Answers:

$\left ( 20.09, 0 \right )$

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

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

$\left ( 5.75,0 \right )$

The graph has no -intercept.

Correct answer:

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

Explanation:

Set f (x) = 0 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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Example Question #1 : Graphing Logarithmic Functions

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

Possible Answers:

y = -7

x = -6

x = -6 and y = -7

x = 6

x = 6 and y = -7

Correct answer:

x = -6

Explanation:

The graph of the logarithmic function

$\ln (x - h) + k$

has as its only asymptote the vertical line

x = h

Here, since h = -6 , the only asymptote is the line

x = -6 .

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Example Question #1 : Graphing Logarithmic Functions

Which is true about the graph of

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

Possible Answers:

All of the answers are correct

None of the answers are correct

The domain of the function is greater than zero

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

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

Correct answer:

All 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.

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Example Question #2 : Graphing Logarithmic Functions

Which of the following is true about the graph of

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

Possible Answers:

The domain is infinite in both directions.

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

It is an odd function.

It is an even function.

The range must be greater than zero.

Correct answer:

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

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 $\small y=x$

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Example Question #81 : Solving And Graphing Logarithms

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

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

Possible Answers:

y = -4

The graph of f(x) does not have a horizontal asymptote.

y = 4

y = -2

y = 2

Correct answer:

The graph of f(x) does not have a horizontal asymptote.

Explanation:

Let $g(x) = \ln x$

In terms of g(x) ,

f(x) =3 g (x+ 4)+ 2

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

The graph of g(x) does not have a horizontal asymptote; therefore, a transformation of this graph, such as that of f(x) , does not have a horizontal asymptote either.

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Example Question #7 : Graphing Logarithmic Functions

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

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

Possible Answers:

x = -2

x = 3

x = 2

x=4

x= -4

Correct answer:

x= -4

Explanation:

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

f(x) =3 g (x+ 4)+ 2 .

The graph of g(x) has as its vertical asymptote the line of the equation x= 0 . The graph of f(x) is the result of three transformations on the graph of g(x) - a left shift of 4 units (+4) , 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 f(x) also shifts left 4 units, to x = -4 .

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #76 : Solving Logarithms

Example Question #211 : Logarithms

Example Question #3121 : Algebra Ii

Example Question #1 : Graphing Logarithmic Functions

Example Question #1 : Graphing Logarithmic Functions

Example Question #1 : Graphing Logarithmic Functions

Example Question #1 : Graphing Logarithmic Functions

Example Question #2 : Graphing Logarithmic Functions

Example Question #81 : Solving And Graphing Logarithms

Example Question #7 : Graphing Logarithmic Functions

All Algebra II Resources

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