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Algebra II : Understanding Logarithms

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Example Questions

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Example Question #21 : Natural Log

Solve: ln(2x-e) = 3

Possible Answers:

$\frac{e^3+e}{2}$

$\frac{(e+1)^3}{2}$

$\frac{e^3-e}{2}$

$\frac{3}{2}e$

$\frac{9-e}{3}$

Correct answer:

$\frac{e^3+e}{2}$

Explanation:

In order to eliminate the natural log, which has a base of , we will need to raise both side as powers of .

$e^{ln(2x-e)} = e^3$

The equation can be simplified to:

2x-e = e^3

Add on both sides.

2x-e +e= e^3+e

2x= e^3+e

Divide by two on both sides.

$\frac{2x}{2}= \frac{e^3+e}{2}$

The answer is: $\frac{e^3+e}{2}$

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Example Question #22 : Natural Log

Try without a calculator:

Which expression is not equivalent to 1?

Possible Answers:

$i^{4}$

$1,000^{\frac{1}{1,000}}$

$\log 10$

$1,000,000^{0}$

$\ln e$

Correct answer:

$1,000^{\frac{1}{1,000}}$

Explanation:

$1,000^{\frac{1}{1,000}}$ is the correct choice.

For all a,N for which the expressions are defined,

$a ^{\frac{1}{N}} = \sqrt[N]{a}$ .

Setting a=1,000, N = 1,000 , this equation becomes

$1,000 ^{\frac{1}{1,000}} = \sqrt[1,000]{1,000}$ - that is, the one thousandth root of 1,000. This is not equal to 1, since if it were, it would hold that $1^{1,000} = 1,000$ - which is not true.

Of the other four expressions:

$\log 10$ , the common, or base ten, logarithm of 10, can be rewritten as $\log_{10}10$ , and $\ln e$ , the natural, or base , logarithm of , can be rewritten as $\log_{e}e$ . A property of logarithms states that for all $N > 0, N \ne 1$ , $\log_{N}N = 1$ . Therefore, $\log 10 = \log_{10}10 = 1$ and $\ln e = \log_{e}e = 1$ .

$1,000,000^{0} = 1$ , since any nonzero number raised to the power of 0 is equal to 1.

By the Power of a Power Property,

$i^{4} = i ^{2 \cdot 2} =( i^{2} )^{2}$

$i^{2} = -1$ , so

$i^{4} = ( -1)^{2}= 1$

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Example Question #1 : Log Base 10

Based on the definition of logarithms, what is ${\log_{10}1000}$ ?

Possible Answers:

100

Correct answer:

Explanation:

For any equation $\log_{10}(y) = x$ , $10^{x } = y$ . Thus, we are trying to determine what power of 10 is 1000. 1000 = 10^3 , so our answer is 3.

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Example Question #2 : Log Base 10

$7 ^{x} = 1,000$

Evaluate .

Possible Answers:

$x = \frac{1,000}{\log 7}$

$x = \frac{3}{\log 7}$

$\log 993$

$x = 3 - \log 7$

$x = 1,000 - \log 7$

Correct answer:

$x = \frac{3}{\log 7}$

Explanation:

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$7 ^{x} = 1,000$

$7 ^{x} = 10^{3}$

$\log 7 ^{x} = \log 10^{3}$

$x \log 7 = 3 \log 10$

$x \log 7 = 3 \cdot 1$

$x \log 7 = 3$

$\frac{x \log 7}{\log 7} = \frac{3}{\log 7}$

$x = \frac{3}{\log 7}$

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Example Question #3 : Log Base 10

What is the value of $\log{10000000}$ ?

Possible Answers:

5.68

7.31

8.13

6.97

Correct answer:

Explanation:

Base-10 logarithms are very easy if the operands are a power of . Begin by rewriting the question:

$\log{10000000}$

Becomes...

$\log{10^7}$

because 10^7=10000000

Applying logarithm rules, you can factor out the :

$7\log{10}$

Now, $\log{10}$ is .

Therefore, your answer is .

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Example Question #4 : Log Base 10

What is the value of $\log{100000}$ ?

Round to the nearest hundreth.

Possible Answers:

4.52

5.01

5.10

Correct answer:

Explanation:

Base-10 logarithms are very easy if the operands are a power of . Begin by rewriting the question:

$\log{100000}$

Becomes...

$\log{10^5}$

because 10^5=100000

Applying logarithm rules, you can factor out the :

$5\log{10}$

Now, $\log{10}$ is .

Therefore, your answer is .

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Example Question #1 : Log Base 10

Many textbooks use the following convention for logarithms:

$log(x) = log_{10}(x)$

ln(x) = log_e(x)

lg(x)=log_2(x)

What is the value of log1,000 ?

Possible Answers:

100

Correct answer:

Explanation:

Remember:

log_ab=c is the same as saying a^c = b .

So when we ask "What is the value of log(1,000) ?", all we're asking is "10 raised to which power equals 1,000?" Or, in an expression:

10^? = 1,000 .

From this, it should be easy to see that log(1000)=3 .

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Example Question #6 : Log Base 10

Evaluate the following expression:

$\small \log (1000)$

Possible Answers:

$\small 3$

$\small 10$

$\small 2$

$\small 5$

$\small 1$

Correct answer:

$\small 3$

Explanation:

Without a subscript a logarithmic expression is base 10.

The expression $\small \log(1000)=\log _{10}(1000)$

The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when

$\small 10^{x}=1000$

We know that $\small 10^{3}=1000$

so $\small \log (1000)=3$

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Example Question #7 : Log Base 10

Assuming the value of is positive, simplify: $\log_{10}10^{3x+2}$

Possible Answers:

-3x-2

30x+20

-30x-20

3x+2

Correct answer:

3x+2

Explanation:

Rewrite the logarithm in division.

$\log_{10}10^{3x+2} = \frac{\log{10}^{3x+2}}{\log 10}$

As a log property, we can pull down the exponent of the power in front as the coefficient.

$\frac{\log{(10)}^{3x+2}}{\log( 10)} = \frac{(3x+2)(\log 10)}{\log (10)}$

Cancel out the $\log(10)$ .

The answer is: 3x+2

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Example Question #8 : Log Base 10

Solve the following:

$\log10000+\log10$

Possible Answers:

$\log10010$

Correct answer:

Explanation:

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\log 10000=x$

is asking:

10^x=10000

x=4

For the second term,

$\log(10)=y$

is asking:

10^y=10

y=1

So, our final answer is 4+1=5

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Algebra II : Understanding Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Natural Log

Example Question #22 : Natural Log

Example Question #1 : Log Base 10

Example Question #2 : Log Base 10

Example Question #3 : Log Base 10

Example Question #4 : Log Base 10

Example Question #1 : Log Base 10

Example Question #6 : Log Base 10

Example Question #7 : Log Base 10

Example Question #8 : Log Base 10

All Algebra II Resources

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