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ISEE Upper Level Math : How to find the solution to an equation

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 #101 : Algebraic Concepts

Find :

$\frac{5^x}{15^{3x}}=8$

Possible Answers:

x=-0.319

x=-0.419

x=-0.219

x=0.319

x=0.219

Correct answer:

x=-0.319

Explanation:

First, simplify the equation as much as possible:

$\frac{5^x}{15^{3x}}=8\Rightarrow \frac{5^x}{(3\times 5)^{3x}}=8$

$\Rightarrow \frac{5^x}{3^{3x}\times 5^{3x}}=8$

$\Rightarrow \frac{5^{(x-3x)}}{3^{3x}}=8$

$\Rightarrow\frac{5^{-2x}}{3^{3x}} =8$

Now we should take log of both sides:

$\frac{5^{-2x}}{3^{3x}} =8\Rightarrow log\ \frac{5^{-2x}}{3^{3x}}=log\ 8$

$\Rightarrow log\ 5^{-2x}-log\ 3^{3x}=log\ 8$

$\Rightarrow (-2x)log5-(3x)log\ 3=log\ 8$

$\Rightarrow (-2x)(0.699)-(3x)(0.477)=0.903$

$\Rightarrow -1.398x-1.431x=0.903$

$\Rightarrow -2.829x=0.903$

$\Rightarrow x=\frac{0.903}{-2.829}=-0.319$

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Example Question #101 : Algebraic Concepts

Solve:

$5(3)^{x+3}=40$

Possible Answers:

x=-2.107

x=1.107

x=2.107

x=-1.107

x=-3.107

Correct answer:

x=-1.107

Explanation:

First we can simplify the equation:

$5(3)^{x+3}=40\Rightarrow (3)^{x+3}=\frac{40}{5}=8$

Now take the log from both sides:

$(3)^{x+3}=8\Rightarrow log\ 3^{x+3}=log\ 8\Rightarrow (x+3)log\ 3=log\ 8$

$\Rightarrow x+3=\frac{log\ 8}{log\ 3}\Rightarrow x=\frac{log\ 8}{log\ 3}-3=1.893-3$

$\Rightarrow x=-1.107$

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Example Question #103 : Algebraic Concepts

Solve the following equation for :

$e^{2x}-4e^x+3=0$

Possible Answers:

x=0 or x=-1.099

x=0 or x=1

x=-1.099 or x=1.099

x=1 or x=-1

x=0 or x=1.099

Correct answer:

x=0 or x=1.099

Explanation:

Let: A=e^x

So we have:

$e^{2x}=e^{(x+x)}=e^x\times x^x=A\times A=A^2$

Now substitute the A=e^x in the equation:

$A^2-4A+3=0\Rightarrow (A-1)(A-3)=0$

$\Rightarrow A-1=0\ or\ A-3=0$

$A-1=0\Rightarrow A=1\Rightarrow e^x=1\Rightarrow ln\ e^x=ln\ 1\Rightarrow x=ln\ 1=0$

$A-3=0\Rightarrow A=3\Rightarrow e^x=3\Rightarrow ln\ e^x=ln\ 3\Rightarrow x=ln\ 3=1.099$

So the answers of the equation are x=0 or x=1.099

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Example Question #102 : Algebraic Concepts

Solve for :

e^x=20

Possible Answers:

x=-2.996

x=1.996

x=2.996

x=-3.996

x=3.996

Correct answer:

x=2.996

Explanation:

Since log is the inverse of the exponential function, we can use to solve this problem:

$e^x=20\Rightarrow ln (e^x)=ln\ 20\Rightarrow x\times lne=ln\ 20$

Use the fact that $ln\ e=1$ :

$x\times 1=ln\ 20\Rightarrow x=2.996$

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Example Question #105 : Algebraic Concepts

Solve the equation for :

$e^{2x}=10+e^0$

Possible Answers:

x=2.199

x=-1.199

x=3.199

x=1.199

x=-2.199

Correct answer:

x=1.199

Explanation:

Any number raised to the power of zero is equal to :

e^0=1

$e^{2x}=10+e^0\Rightarrow e^{2x}=10+1=11\Rightarrow ln(e^{2x})=ln\ 11$

$\Rightarrow 2xln\ e=ln\ 11\Rightarrow 2x=ln\ 11$

$\Rightarrow x=\frac{ln\ 11}{2}=\frac{2.398}{2}=1.199$

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Example Question #801 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Solve:

$15e^{0.5x}=200$

Possible Answers:

x=-1.18

x=1.18

x=-5.18

x=5.18

x=3.18

Correct answer:

x=5.18

Explanation:

First, simplify the equation by dividing both sides by :

$15e^{0.5x}=200\Rightarrow e^{0.5x}=\frac{200}{15}=\frac{40}{3}$

Now we can take of both sides:

$e^{0.5x}=\frac{40}{3}\Rightarrow ln\ e^{0.5x}=ln\ (\frac{40}{3})$

$\Rightarrow 0.5x=2.59$

$\Rightarrow x=\frac{2.59}{0.5}=5.18$

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Example Question #107 : Algebraic Concepts

Solve for :

$e^{(2x+2)}=100$

Possible Answers:

x=1.203

x=1.403

x=1.103

x=1.303

x=-1.303

Correct answer:

x=1.303

Explanation:

Take the of both sides:

$e^{(2x+2)}=100\Rightarrow ln\ e^{(2x+2)}=ln\ 100$

$\Rightarrow 2x+2=4.605$

$\Rightarrow 2x=4.605-2=2.605$

$\Rightarrow x=\frac{2.605}{2}=1.303$

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Example Question #108 : Algebraic Concepts

Solve the following equation for :

$e^{2x}-2e^x+1=0$

Possible Answers:

x=-1

x=1

x=2

x=-2

x=0

Correct answer:

x=0

Explanation:

Let A=e^x .

$e^{2x}=e^{(x+x)}=e^x\times x^x=A\times A=A^2$

Now substitute A=e^x in the equation:

$A^2-2A+1=0\Rightarrow (A-1)^2=0\Rightarrow A-1=0\Rightarrow A=1$

$A=1\Rightarrow e^x=1$

Take the of both sides:

$e^x=1\Rightarrow ln\ e^x=ln\ 1\Rightarrow x=ln\ 1$

$x=ln\ 1=0$

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Example Question #109 : Algebraic Concepts

Solve the equation for :

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

Possible Answers:

x=-0.549

x=-0.6

x=0.6

x=0.458

x=0.549

Correct answer:

x=0.549

Explanation:

First, simplify the equation as much as possible:

$\frac{e^{3x}}{e^x}=3\Rightarrow e^{(3x-x)}=3\Rightarrow e^{2x}=3$

Now we can take the of both sides:

$e^{2x}=3\Rightarrow ln\ e^{2x}=ln\ 3$

$\Rightarrow 2x=ln\ 3$

$\Rightarrow x=\frac{ln\ 3}{2}=\frac{1.098}{2}$

$\Rightarrow x=0.549$

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Example Question #110 : How To Find The Solution To An Equation

Solve the following equation for :

$\frac{e^{4x}+1}{e^{4x}-1}=4$

Possible Answers:

x=0.654

x=1

x=0.127

x=0.327

x=0.227

Correct answer:

x=0.127

Explanation:

First,simplify the equation as much as possible:

$\frac{e^{4x}+1}{e^{4x}-1}=4\Rightarrow e^{4x}+1=4(e^{4x}-1)$

$\Rightarrow e^{4x}+1=4e^{4x}-4$

$\Rightarrow 4e^{4x}-e^{4x}=1+4$

$\Rightarrow 3e^{4x}=5$

Now take of both sides

$3e^{4x}=5\Rightarrow ln\ (3e^{4x})=ln\ 5$

$\Rightarrow ln\ 3+ln\ e^{4x}=ln\ 5$

$\Rightarrow ln\ e^{4x}=ln\5-ln\ 3$

$\Rightarrow 4x=1.609-1.099$

$\Rightarrow x=\frac{0.51}{4}=0.127$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to find the solution to an equation

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #101 : Algebraic Concepts

Example Question #101 : Algebraic Concepts

Example Question #103 : Algebraic Concepts

Example Question #102 : Algebraic Concepts

Example Question #105 : Algebraic Concepts

Example Question #801 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #107 : Algebraic Concepts

Example Question #108 : Algebraic Concepts

Example Question #109 : Algebraic Concepts

Example Question #110 : How To Find The Solution To An Equation

All ISEE Upper Level Math Resources

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