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Algebra II : Simplifying Logarithms

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Example Questions

Example Question #1 : Logarithms With Exponents

Evaluate the following expression

$\small \log(1000^{2})$

Possible Answers:

$\small \frac{\log (2)}{\log (10)}$

$\small 9$

$\small 3$

$\small 6$

$\small \frac{1}{9}$

Correct answer:

$\small 6$

Explanation:

Since the exponent is inside the parentheses, you must take the square of 1000 before finding the logarithim. Therefore

$\small \log(1000^{2})=\log (1000000)=6$

because $\small 10^{6}=1,000,000$

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Example Question #3 : Logarithms With Exponents

Evaluate the following expression

$\small \log _{3}(9^{2})$

Possible Answers:

$\small 81$

$\small 1/4$

$\small 4$

$\small 2$

$\small 27$

Correct answer:

$\small 4$

Explanation:

Since the exponent is inside the parentheses, you must determine the value of the exponential expression first.

$\small 9^{2}=81$

then you solve the logarithm

$\small \log _{3}(81)=4$ because $\small 3^{4}=81$

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Example Question #4 : Logarithms With Exponents

Evaluate the following for all integers of $\small a$ and $\small b$

$\small a^{\log _{a}(b)}$

Possible Answers:

$\small b$

$\small a$

$\small \frac{a}{b}$

$\small \log_{a} (b)$

$\small \log _{b}(a)$

Correct answer:

$\small b$

Explanation:

$\small \log _{a}(b)$ gives us the exponent to which $\small a$ must be raised to yield $\small b$

When $\small a$ is actually raised to that number in the equation given, the answer must be $\small b$

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Example Question #5 : Logarithms With Exponents

Evaluate the following expression

$\left ( \log1000\right)^{2}=$

Possible Answers:

$\small 9$

$\small \log (2000)$

$\small 6$

$\small 3$

$\small \frac{1}{1000}$

Correct answer:

$\small 9$

Explanation:

This is a simple exponent of a logarithmic answer.

$\small \log(1000)=3$ because $\small 10^{3}=1000$

$\small 3^{2}=9$

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Example Question #6 : Logarithms With Exponents

Evaluate the following expression

$\left( \log _{2}16 \right)^{3}$

Possible Answers:

$\small 48$

$\small 12$

$\small 4$

$\small 1024$

$\small 64$

Correct answer:

$\small 64$

Explanation:

This is a two step problem. First find the log base 2 of 16

$\small \log _{2}(16)=4$ because $\small 2^{4}=16$

then

$\small 4^{3}=64$

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Example Question #2 : Logarithms With Exponents

Which of the following equations is valid?

Possible Answers:

$\small a^{\log _{a}(b)}=a$

none of the other answers are correct

$\small \log _{a}(a^{b})=b$

$\small \log_{2b}(a)= \log_{b}(a)^{2}$

$\small \log (a)^{2}=\log2a$

Correct answer:

$\small \log _{a}(a^{b})=b$

Explanation:

Since a logarithm answers the question of which exponent to raise the base to receive the number in parentheses, if the number in parentheses is the base raised to an exponent, the exponent must be the answer.

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Example Question #61 : Simplifying Logarithms

Rewrite the following logarithmic expression into expanded form (that is, using addition and/or subtraction):

$\small \ln{(AB)}^{4}$

Possible Answers:

$\small \ln{A^{4}}+\ln{B^4}$

$\small 4\ln{AB}$

$\small 4\ln{A}+4\ln{B}$

$\small 4\ln{AB}$

$\small 4\ln{A}+\ln{B}$

Correct answer:

$\small 4\ln{A}+4\ln{B}$

Explanation:

Before we do anything, the exponent of 4 must be moved to the front of the expression, as the rules of logarithms dictate. We end up with $\small 4\ln{AB}$ . Remember that a product inside of a logarithm can be rewritten as a sum: $\small 4(\ln{A}+\ln{B})$ . Distributing, we get $\small 4\ln{A}+4\ln{B}$ .

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Example Question #11 : Logarithms With Exponents

Use

$\log4=0.602$

and

$\log12=1.079$

Evaluate:

$\log64$

Possible Answers:

1.806

0.218

1.204

0.477

1.681

Correct answer:

1.806

Explanation:

Since the question gives,

$\log4=0.602$

and

$\log12=1.079$

To evaluate

$\log64$

manipulate the expression to use what is given.

$=\log4^3$

$=3\log4$

=3*0.602

=1.806

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Example Question #380 : Mathematical Relationships And Basic Graphs

Simplify: $2e^{ln(2x+1)}$

Possible Answers:

4x+2

4x+1

$x+\frac{1}{2}$

$2^{2x+1}$

4x^2+4x+1

Correct answer:

4x+2

Explanation:

According to log rules, when an exponential is raised to the power of a logarithm, the exponential and log will cancel out, leaving only the power.

Simplify the given expression.

$2e^{ln(2x+1)} = 2(2x+1)$

Distribute the integer to both terms of the binomial.

The answer is: 4x+2

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Example Question #12 : Logarithms With Exponents

Simplify: $5ln(e^{5(5-x)})$

Possible Answers:

25-x

5-x

125-5x

125-25x

x-5

Correct answer:

125-25x

Explanation:

The natural log has a default base of .

This means that the expression written can also be:

$5ln_e(e^{5(5-x)})$

Recall the log property that:

ln_k(k^n) = n

This would eliminate both the natural log and the base, leaving only the exponent.

The natural log and the base will be eliminated.

The expression will simplify to:

5[5(5-x)] = 5[25-5x] = 125-25x

The answer is: 125-25x

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Algebra II : Simplifying Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Logarithms With Exponents

Example Question #3 : Logarithms With Exponents

Example Question #4 : Logarithms With Exponents

Example Question #5 : Logarithms With Exponents

Example Question #6 : Logarithms With Exponents

Example Question #2 : Logarithms With Exponents

Example Question #61 : Simplifying Logarithms

Example Question #11 : Logarithms With Exponents

Example Question #380 : Mathematical Relationships And Basic Graphs

Example Question #12 : Logarithms With Exponents

All Algebra II Resources

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