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Algebra II : Logarithms and exponents

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Example Questions

Example Question #41 : Logarithms

Which equation is equivalent to:

$\log _{4}\left( \frac{1}{2}\right)=x$

Possible Answers:

$(4)^{x}=\left(\frac{1}{2}\right)^{x}$

$4^{\frac{1}{2}}=x$

$\frac{1^{4}}{2}=x^{4}$

$4^{x}=\frac{1}{2}$

$\left(\frac{1}{2}\right)^{4}=x$

Correct answer:

$4^{x}=\frac{1}{2}$

Explanation:

$log{_{a}}^{b}=x$ , b>0

$\Rightarrow a^{x}=b$

So, ${{log_{4}}^{\frac{1}{2}}}=x\Rightarrow 4^{x}=\frac{1}{2}$

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Example Question #3 : Exponential And Logarithmic Functions

$Log_{b}x=y$

What is the inverse of the log function?

Possible Answers:

$y^{b}=x$

$x^{b}=y$

$b=x\cdot y$

$b^{x}=y$

$b^{y}=x$

Correct answer:

$b^{y}=x$

Explanation:

This is a general formula that you should memorize. The inverse of $Log_{b}x=y$ is $b^{y}=x$ . You can use this formula to change an equation from a log function to an exponential function.

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Example Question #42 : Understanding Logarithms

Solve for :

7^b=54

Round to the nearest hundredth.

Possible Answers:

3.21

2.05

Cannot be computed

1.45

0.49

Correct answer:

2.05

Explanation:

To solve this, you need to set up a logarithm. Our exponent is . The logarithm's base is . The value is the operand of the logarithm. Therefore, we can write an equation:

$\log_7{54} = b$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_7{54}$

to...

$\frac{\log{54}}{\log{7}}$

You can put this into your calculator and get:

2.04993228926876 , or rounded, 2.05

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Example Question #3 : Logarithms And Exponents

Solve for :

31^r=842

Round to the nearest hundredth.

Possible Answers:

1.73

1.15

0.51

2.31

1.96

Correct answer:

1.96

Explanation:

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of (31) is the base. This is the logarithm's base. The value 842 is the operand of the logarithm. Therefore, we can write an equation:

$\log_3_1{842} = r$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_3_1{842}$

to...

$\frac{\log{842}}{\log{31}}$

You can put this into your calculator and get:

1.96150410969636 , or rounded, 1.96

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Example Question #4 : Logarithms And Exponents

Solve for :

4^x=6317

Round to the nearest hundredth.

Possible Answers:

3.24

5.65

0.16

2.81

6.31

Correct answer:

6.31

Explanation:

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of (4) is the base. This is the logarithm's base. The value 842 is the operand of the logarithm. Therefore, we can write an equation:

$\log_4 {6317} = x$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_4 {6317}$

to...

$\frac{\log{6317}}{\log{4}}$

You can put this into your calculator and get:

6.3125119284402 , or rounded, 6.31

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Example Question #1 : Logarithms And Exponents

Solve for :

5^t=40150

Round to the nearest hundredth.

Possible Answers:

6.41

4.14

6.59

5.91

0.16

Correct answer:

6.59

Explanation:

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of (5) is the base. This is the logarithm's base. The value 40150 is the operand of the logarithm. Therefore, we can write an equation:

$\log_5 {40150} = t$

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

$\log_5 {40150}$

to...

$\frac{\log{40150}}{\log{5}}$

You can put this into your calculator and get:

6.58638499657496 , or rounded, 6.59

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Example Question #7 : Logarithms And Exponents

Write the equation $3^{-4}=\frac{1}{81}$ in logarithmic form.

Possible Answers:

$log_3\frac{1}{81}=-4$

$log_3{-4}=\frac{1}{81}$

$log_\frac{1}{81}3=-4$

$log_{-4}\frac{1}{81}=3$

$log_{-4}3=\frac{1}{81}$

Correct answer:

$log_3\frac{1}{81}=-4$

Explanation:

For logarithmic equations, a^c=b can be rewritten as log_ab=c .

In this expression, is the base of the equation (). is the exponent () and $\frac{1}{81}$ is the term ().

In putting each term in its appropriate spot, the exponential equation can be converted to $log_3\frac{1}{81}=-4$ .

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Example Question #8 : Logarithms And Exponents

Solve the following logarithm for :

log_5x=3

Possible Answers:

x=15

x=25

x=3

x=125

Correct answer:

x=125

Explanation:

Solve the following logarithm:

log_5x=3

Recall that we can convert logarithms to exponential form via the following:

log_ba=c b^c=a

Using this approach, convert the given log to exponential form:

5^3=x

x=125

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Example Question #4 : Exponential And Logarithmic Functions

Rewrite the following expression as an exponential expression:

log_3b=14,000

Possible Answers:

b^3=14,000

$3^{14,000}=b$

14,000^3=b

$b^{14,000}=3$

Correct answer:

$3^{14,000}=b$

Explanation:

Rewrite the following expression as an exponential expression:

log_3b=14,000

Recall the following property of logs and exponents:

log_b(a)=c

Can be rewritten in the following form:

b^c=a

So, taking the log we are given;

log_3b=14,000

We can rewrite it in the form:

$3^{14,000}=b$

So b must be a really huge number!

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Example Question #5 : Exponential And Logarithmic Functions

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Possible Answers:

147^b=13

$b^{13}=147$

$13^{b}=147$

$13^{147}=b$

Correct answer:

$13^{147}=b$

Explanation:

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Recall the following:

This

log_b(a)=c

Can be rewritten as

b^c=a

So, our given logarithm

$log_{13}b=147$

Can be rewritten as

$13^{147}=b$

Fortunately we don't need to expand, because this woud be a very large number!

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Algebra II : Logarithms and exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #41 : Logarithms

Example Question #3 : Exponential And Logarithmic Functions

Example Question #42 : Understanding Logarithms

Example Question #3 : Logarithms And Exponents

Example Question #4 : Logarithms And Exponents

Example Question #1 : Logarithms And Exponents

Example Question #7 : Logarithms And Exponents

Example Question #8 : Logarithms And Exponents

Example Question #4 : Exponential And Logarithmic Functions

Example Question #5 : Exponential And Logarithmic Functions

All Algebra II Resources

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