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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #74 : Solving Logarithms

Solve log_2(x-4) = 3-log_2 (x+3) /

Possible Answers:

Correct answer:

Explanation:

The first thing we can do is move both log functions to one side of the equation:

log_2(x-4) + log_2 (x+3)= 3

Then we can combine the log functions (remember, when you add logs, we multiply the terms inside):

log_2((x-4)(x+3))= 3

Now we can rewrite the equation in exponent form (and FOIL the multiplied terms):

x^2-x-12=2^3

We can collect all the terms on one side of the equation, and then solve the quadratic:

x^2-x-20=0

(x-5)(x+4)=0

x=-4, 5

However, if we plug into the initial equation, we would be taking the log of a negative number, which we can't do, so it's not a valid solution:

x=5

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Example Question #75 : Solving Logarithms

Solve log_7(6x-16)=log_7(x-1) .

Possible Answers:

Correct answer:

Explanation:

We first put both logs on one side of the equation:

log_7(6x-16)-log_7(x-1)=0

Now we combine the log terms (remember, when we subtract logs we divide the terms inside):

$log_7(\frac{(6x-16)}{(x-1)})=0$

We can now rewrite the equation in exponential form:

$\frac{6x-16}{x-1}=7^0$

Anything raised to the power is , and now we can solve algebraically:

6x-16=x-1

5x=15

x=3

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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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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #74 : Solving Logarithms

Example Question #75 : Solving Logarithms

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

All Algebra II Resources

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