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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #16 : Solving Logarithms

Solve for $\small x$ :

$\small \log_{2}(x+6)=4$

Possible Answers:

$\textup{No solution}$

$\frac{1}{4}$

Correct answer:

Explanation:

Logarithms are another way of writing exponents. In the general case, $\small \log_a{x}=b$ really just means $\small a^{b}=x$ . We take the base of the logarithm (in our case, 2), raise it to whatever is on the other side of the equal sign (in our case, 4) and set that equal to what is inside the parentheses of the logarithm (in our case, x+6). Translating, we convert our original logarithm equation into $\small 2^{4}=x+6$ . The left side of the equation yields 16, thus $\small x=10$ .

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Example Question #401 : Mathematical Relationships And Basic Graphs

$\log_{5}x=3$

Possible Answers:

$\frac{1}{25}$

125

$\frac{1}{125}$

Correct answer:

125

Explanation:

To solve this equation, remember log rules

$\log_{b}x=y; b^{y}=x$ .

This rule can be applied here so that

5^3=x

and

125=x.

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Example Question #11 : Solving And Graphing Logarithms

$\log_{4}64-\log_{3}81=$

Possible Answers:

Correct answer:

Explanation:

To solve this equation, you must first simplify the log expressions and remember log laws ( $\log_{b}x=y$ ; $b^{y}=x$ ). Therefore, for the first expression, $4^{x}=64$ so x=3 . For the second expression, 3^x=81 so x=4. Then, substitute those numbers in so that you get 3-4=-1 .

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Example Question #19 : Solving Logarithms

Solve for

$log_{5}(x) = 2$

Possible Answers:

x = 12

x = 25

x = 32

x = 15

x = 5

Correct answer:

x = 25

Explanation:

$log_{5}(x) = 2$

Remember that a logarithm is nothing more than an exponent. The equation reads, the logarithm (exponent) with base that gives is .

x = 5^2

x = 25

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Example Question #20 : Solving Logarithms

Solve the equation

$\log_4(4x+7)=\log_4(11x)$

Possible Answers:

None of the other answers

x=2

x=3

x=0.5

x=1

Correct answer:

x=1

Explanation:

Since both sides are a log with the same base, their arguments must be equal.

Setting them equal:

4x+7=11x

Solve for x:

7=7x

1=x

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Example Question #21 : Solving Logarithms

Solve the equation

$\ln(3x-2)=\ln(6x)$

Possible Answers:

x=1

No Solution

$x=\frac{2}{3}$

x=2

$x=\frac{-2}{3}$

Correct answer:

No Solution

Explanation:

Set the arguments of the natural log equal and solve for

3x-2=6x

-2=3x

$\frac{-2}{3}=x$

However, you cannot take the natural log of a negative number. Therefore; there is No Solution

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Example Question #22 : Solving Logarithms

Solve:

$\log_{2}24$

Possible Answers:

$\log24$

$\log12$

$\frac{\log24}{\log2}$

$-\frac{\log24}{\log2}$

Correct answer:

$\frac{\log24}{\log2}$

Explanation:

To solve this problem, it is easiest to start by thinking about what the logarithm means:

2^x=24

Now, we take the log of both sides (either base-10 or natural):

$\log(2^x)=\log24$

Next, using the properties of logarithms, we can bring the exponent x in front of the logarithm:

$x\log2=\log24$

Finally, solve for x:

$x=\frac{\log24}{\log2}$

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Example Question #23 : Solving Logarithms

Evaluate

$\log_5 25 = x$

Possible Answers:

Correct answer:

Explanation:

Logarithms are inverses of exponents.

$\log_5 25$ is asking how many fives multiplied together are equal to 25.

The formula to solve logarithmic equations is as follows.

$log_ab=c\Rightarrow a^c=b$

Applying this formula to our particular problem results in the following.

$\log_5 25\Rightarrow 5^x=25$

5^2 = 5*5 = 25

So x= 2 .

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Example Question #402 : Mathematical Relationships And Basic Graphs

Solve:

$\log_84+\log_810$

Possible Answers:

$\log32$

$\frac{\log40}{\log8}$

$\log5$

$\frac{\log14}{\log8}$

Correct answer:

$\frac{\log40}{\log8}$

Explanation:

To solve, we must keep in mind that a logarithm is asking the following:

$\log_ab$ means a^x=b , and we must find x.

First, we can rewrite the sum of the base-8 logarithms as a product (this is a property of logarithms):

$\log_8(4\cdot 10)=\log_840$

Now, we use the above formula to find what this numerically equals:

8^x=40

Take the natural or common logarithm of both sides, which allows us to bring the power x in front of the logarithm:

$\log8^x=\log40$

$x\log8=\log40$

$x=\frac{\log40}{\log8}$

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Example Question #25 : Solving Logarithms

Solve for x:

$4^x=3^{x^2}$

Possible Answers:

$\log\frac{3}{4}$

$\frac{\log4}{\log3}$

$\frac{\log3}{\log4}$

$\log1$

$\log\frac{4}{3}$

Correct answer:

$\frac{\log4}{\log3}$

Explanation:

To solve for x, we must take the logarithm of both sides (common or natural, it doesn't matter):

$\log(4^x)=\log(3^{x^2})$

In doing this, we can now bring the exponents in front of the logarithms:

$x\log4=x^2\cdot\log3$

Now, solve for x:

$\log4=x\log3$

$x=\frac{\log4}{\log3}$

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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 #16 : Solving Logarithms

Example Question #401 : Mathematical Relationships And Basic Graphs

Example Question #11 : Solving And Graphing Logarithms

Example Question #19 : Solving Logarithms

Example Question #20 : Solving Logarithms

Example Question #21 : Solving Logarithms

Example Question #22 : Solving Logarithms

Example Question #23 : Solving Logarithms

Example Question #402 : Mathematical Relationships And Basic Graphs

Example Question #25 : Solving Logarithms

All Algebra II Resources

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