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SAT II Math I : Exponents and Logarithms

Study concepts, example questions & explanations for SAT II Math I

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Example Question #1 : Mathematical Relationships

How many elements are in a set that has exactly 128 subsets?

Possible Answers:

None of the other responses is correct.

Correct answer:

Explanation:

A set with elements has $2 ^{N}$ subsets.

Solve:

$2 ^{N} = 128$

$\ln 2 ^{N} = \ln 128$

$N \ln 2 = \ln 128$

$N = \frac{\ln 128}{\ln 2} = \frac{4.8520 }{0.6931} = 7$

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Example Question #2 : Exponents And Logarithms

Rewrite as a single logarithmic expression:

$\ln x - 2 \ln (x + 2)$

Possible Answers:

$\ln \frac{1}{x^{2} +3x +4}$

$\ln \frac{x}{x^{2} +4x + 4}$

$\ln \frac{1}{x +4}$

$\ln \left (-x^{2} -3x -4 \right )$

$\ln \frac{x}{2x + 4}$

Correct answer:

$\ln \frac{x}{x^{2} +4x + 4}$

Explanation:

Using the properties of logarithms

$n \ln a = \ln a^{n}$ and $\ln a - \ln b = \ln \frac{a}{b}$ ,

we simplify as follows:

$\ln x - 2 \ln (x + 2)$

$= \ln x - \ln (x + 2)^{2}$

$= \ln x - \ln (x^{2} +4x + 4)$

$= \ln \frac{x}{x^{2} +4x + 4}$

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Example Question #1 : Mathematical Relationships

Solve: $log_{3}27$

Possible Answers:

$\frac{1}{9}$

Correct answer:

Explanation:

In order to solve this problem, covert 27 to the correct base and power.

$log_{3}27 = log_{3}\: 3^3$

Since log_xx^n = n , the correct answer is .

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Example Question #1 : Exponents And Logarithms

Simplify $(x^{-4})^7$

Possible Answers:

$x^{-28}$

x^3

$x^{-11}$

$x^{\frac{7}{4}}$

$x^{-47}$

Correct answer:

$x^{-28}$

Explanation:

When an exponent is raised by another exponent, we just multiply the powers.

$(x^{-4})^7=x^{-4*7}=x^{-28}$

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Example Question #1 : Exponents And Logarithms

Simplify:

3^9+3^9+3^9

Possible Answers:

$27^{9}$

$3^{10}$

$9^{27}$

9^9

$3^{27}$

Correct answer:

$3^{10}$

Explanation:

When adding exponents, we don't add the exponents or multiply out the bases. Our goal is to see if we can factor anything. We do see three 3^9 . Let's factor.

$3^9+3^9+3^9=3^9(1+1+1)=3^9(3)=3^{10}$ Remember when multiplying exponents, we just add the powers.

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Example Question #1 : Exponents And Logarithms

Solve and simplify.

$\sqrt[3]{125}$

Possible Answers:

$5\sqrt{5}$

$5\sqrt[3]{5}$

Correct answer:

Explanation:

$\sqrt[3]{125}$ Another way to write this is x^3=125 . The only number that makes 125 is 5^3 .

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Example Question #7 : Exponents And Logarithms

Simplify:

$\log_3729$

Possible Answers:

$3^{729}$

3^9

243

Correct answer:

Explanation:

$\log_3729$ is the same as 3^x=729 . Let's factor out 729 . It's the same as 3^6 . Therefore x=6 which is the answer to our question.

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Example Question #2 : Exponents And Logarithms

Simplify:

$\log50-\log5$

Possible Answers:

$\log250$

$\log45$

$5\log50$

Correct answer:

Explanation:

When dealing with subtraction in regards to logarithms, it's the same as dividing the numbers.

$\log50-\log5=\frac{\log50}{\log5}=\log \frac{50}{5}=\log10=1$

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Example Question #3 : Exponents And Logarithms

Simplify:

$\log_618+\log_672$

Possible Answers:

$3\log_618$

$6\log90$

$\log_690$

Correct answer:

Explanation:

When dealing with addition in regards to logarithms, it's the same as multiplying the numbers.

$\log_618+\log_672=\log_6(18*72)=\log_61296=4$

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Example Question #10 : Exponents And Logarithms

Solve: $(x^{2})^{5}$ when x=2 .

Possible Answers:

1024

100

1000

Correct answer:

1024

Explanation:

Power rule says when an exponent is raised to another exponent, you must multiply the exponents.

So 2*5=10 and our expression is now $x^{10}$ .

Plug in the given value to get

$2^{10}=1024$ .

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SAT II Math I : Exponents and Logarithms

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Mathematical Relationships

Example Question #2 : Exponents And Logarithms

Example Question #1 : Mathematical Relationships

Example Question #1 : Exponents And Logarithms

Example Question #1 : Exponents And Logarithms

Example Question #1 : Exponents And Logarithms

Example Question #7 : Exponents And Logarithms

Example Question #2 : Exponents And Logarithms

Example Question #3 : Exponents And Logarithms

Example Question #10 : Exponents And Logarithms

All SAT II Math I Resources

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