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ISEE Upper Level Math : Equations

Study concepts, example questions & explanations for ISEE Upper Level Math

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #91 : Equations

If 2^x=8^y , find the ratio of to .

Possible Answers:

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

$\frac{x}{y}=-1$

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

$\frac{x}{y}=1$

$\frac{x}{y}=4$

Correct answer:

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

Explanation:

We know that 8=2^3 , so we can write:

$2^x=8^y\Rightarrow 2^x=(2^3)^{y}\Rightarrow 2^x=2^{3y}$

Since the bases are the same, set the exponents equal to one another:

$x=3y\Rightarrow \frac{x}{y}=3$

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Example Question #92 : Equations

If 2^x=7^y , give the ratio of to .

Possible Answers:

$\frac{x}{y}=2.8$

$\frac{x}{y}=4$

$\frac{x}{y}=3.8$

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

$\frac{x}{y}=4.8$

Correct answer:

$\frac{x}{y}=2.8$

Explanation:

In this problem, and cannot be written as powers of the same integer, so we should use logarithms:

$2^x=7^y\Rightarrow log\ 2^x=log \7^y$

$\Rightarrow xlog\ 2=ylog\ 7$

$\Rightarrow \frac{x}{y}=\frac{log\ 7}{log\ 2} =2.8$

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Example Question #93 : Equations

If $3^{4x-4}=81^{y-1}$ , give the ratio of to .

Possible Answers:

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

$\frac{x}{y}=-1$

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

$\frac{x}{y}=1$

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

Correct answer:

$\frac{x}{y}=1$

Explanation:

In this problem, can be written as the power of :

81=3^4

$3^{4x-4}=81^{y-1}\Rightarrow 3^{4x-4}=(3^4)^{y-1}$

$\Rightarrow 3^{4x-4}=3^{4(y-1)}\Rightarrow 3^{4x-4}=3^{4y-4}$

Since the bases are the same, set the exponents equal to one another:

$4x-4=4y-4\Rightarrow 4x-4+4=4y-4+4$

$\Rightarrow 4x=4y$

$\Rightarrow \frac{x}{y}=\frac{4}{4}=1$

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Example Question #94 : Equations

Solve the following equation:

$2^{x-1}=7^{x+3}$

Possible Answers:

x=7.213

x=-6.213

x=5.213

x=6.213

x=-5.213

Correct answer:

x=-5.213

Explanation:

In this problem, and cannot be written as powers of the same integer, so we should use logarithms:

$2^{x-1}=7^{x+3}\Rightarrow log\ 2^{x-1}=log\ 7^{x+3}$

$\Rightarrow (x-1)log\ 2=(x+3)\ log 7$

$\Rightarrow (x-1)\times 0.301=(x+3)\times 0.845$

$\Rightarrow 0.301x-0.301=0.845x+3\times 0.845$

$\Rightarrow 0.301x-0.845x=2.535+0.301$

$\Rightarrow -0.544x=2.836$

$\Rightarrow x=\frac{2.836}{-0.544}=-5.213$

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Example Question #95 : Equations

Solve the following equation:

$5^{x}=2^{x-1}$

Possible Answers:

x=0.656

x=-0.656

x=0.756

x=-0.856

x=-0.756

Correct answer:

x=-0.756

Explanation:

In this problem, and cannot be written as powers of the same integer, so we should use logarithms:

$5^{x}=2^{x-1}\Rightarrow log\ 5^{x}=log\ 2^{x-1}$

$\Rightarrow xlog\ 5=(x-1)\ log 2$

$\Rightarrow x\times 0.699=(x-1)\times 0.301$

$\Rightarrow 0.699x=0.301x-0.301$

$\Rightarrow 0.699x-0.301x=-0.301$

$\Rightarrow 0.398x=-0.301$

$\Rightarrow x=\frac{-0.301}{0.398}=-0.756$

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Example Question #96 : Equations

Solve for :

5^x=8

Possible Answers:

x=1.492

x=1.392

x=1.192

x=1.292

x=1.592

Correct answer:

x=1.292

Explanation:

In this problem, and cannot be written as powers of the same integer, so we should take the log of both sides:

$5^x=8\Rightarrow log\ 5^x=log\ 8\Rightarrow xlog\ 5=log8$

$\Rightarrow x=\frac{log\ 8}{log\ 5}$

$\Rightarrow x=1.292$

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Example Question #97 : Equations

Solve the following equation:

4^x=128

Possible Answers:

x=3.5

x=4.5

x=2

x=2.5

x=3

Correct answer:

x=3.5

Explanation:

We know that 4=2^2 and 128=2^7 .

$4^x=128\Rightarrow \left [(2)^2 \right ]^x=2^7\Rightarrow 2^{2x}=2^7$

Since the bases are the same, set the exponents equal to one another:

$2x=7\Rightarrow x=\frac{7}{2}=3.5$

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Example Question #98 : Algebraic Concepts

Solve the equation for :

$3^{3x}=80$

Possible Answers:

x=1.23

x=1.38

x=1.33

x=1.43

x=1.45

Correct answer:

x=1.33

Explanation:

In this problem, and cannot be written as powers of the same integer, so we should use logarithms:

$3^{3x}=80\Rightarrow log\ 3^{3x}=log\ 80$

$\Rightarrow 3x log3=log\ 80$

$\Rightarrow x=\frac{1}{3}\frac{log\ 80}{log3}=\frac{1}{3}\times 3.989$

$\Rightarrow x=1.33$

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Example Question #99 : Algebraic Concepts

Find :

$(\frac{1}{2})^x=5$

Possible Answers:

x=1.322

x=2.322

x=-2.322

x=3.322

x=-1.322

Correct answer:

x=-2.322

Explanation:

We need to take the log from both sides:

$(\frac{1}{2})^x=5\Rightarrow log\ (\frac{1}{2})^x=log\ 5$

$\Rightarrow xlog\ (\frac{1}{2})=log\ 5$

$\Rightarrow x=\frac{log\ 5}{log\ (\frac{1}{2})}=-2.322$

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Example Question #98 : Equations

Find :

$(\frac{3}{4})^{2x-1}=7$

Possible Answers:

x=2.882

x=-3.882

x=-1.882

x=-2.882

x=3.882

Correct answer:

x=-2.882

Explanation:

We need to take the log from both sides:

$(\frac{3}{4})^{2x-1}=7\Rightarrow log\ (\frac{3}{4})^{2x-1}=log\ 7$

$\Rightarrow (2x-1)log\ (\frac{3}{4})=log\ 7$

$\Rightarrow 2x-1=\frac{log\ 7}{log\ \frac{3}{4}}$

$\Rightarrow 2x-1=-6.764$

$\Rightarrow 2x=1-6.764$

$\Rightarrow x=\frac{1-6.764}{2}=\frac{-5.764}{2}$

$\Rightarrow x=-2.882$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Equations

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #91 : Equations

Example Question #92 : Equations

Example Question #93 : Equations

Example Question #94 : Equations

Example Question #95 : Equations

Example Question #96 : Equations

Example Question #97 : Equations

Example Question #98 : Algebraic Concepts

Example Question #99 : Algebraic Concepts

Example Question #98 : Equations

All ISEE Upper Level Math Resources

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