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ISEE Upper Level Math : Equations

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Example Questions

Example Question #111 : Algebraic Concepts

Solve the following equation for :

$3^{x+2}-4=12$

Possible Answers:

x=0.424

x=0.324

x=0.624

x=0.524

x=0.224

Correct answer:

x=0.524

Explanation:

First, simplify the equation:

$3^{x+2}-4=12\Rightarrow 3^{x+2}=12+4=16$

Next take the log of sides:

$3^{x+2}=16\Rightarrow log \ 3^{x+2}=log\ 16$

$\Rightarrow (x+2)log\ 3=log\ 16$

$\Rightarrow x+2=\frac{log\ 16}{log\ 3}$

$\Rightarrow x=\frac{log\ 16}{log\ 3}-2=2.524-2$

$\Rightarrow x=0.524$

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Example Question #111 : Equations

Find :

$log_{x}^{64}=3$

Possible Answers:

x=-4

x=2

x=3

x=1

x=4

Correct answer:

x=4

Explanation:

$log_{x}^{64}=3\Rightarrow 64=x^3\Rightarrow x=\sqrt[3]{64}\Rightarrow x=4$

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Example Question #112 : Equations

Find :

$log_{x}^{64}=log_{3}^{36}-log_{3}^{4}$

Possible Answers:

x=8

$x=8\ or \ x=-8$

x=4

x=-8

x=-4

Correct answer:

x=8

Explanation:

Since the two logarithmic expressions of the right side of the equation have the same base, we can use the quotient rule:

$log_{x}^{64}=log_{3}^{36}-log_{3}^{4}\Rightarrow log_{x}^{64}=log_{3}^{\frac{36}{4}}\Rightarrow log_{x}^{64}=log_{3}^{9}$

We know that 9=3^2 .

$log_{3}^{9}= log_{3}^{3^{(2)}}=2log_{3}^{3}=2$

Then we can write:

$log_{x}^{64}=2\Rightarrow 64=x^2\Rightarrow x=\sqrt{64}=\pm 8$

Logarithmic equations cannot have a negative base, so x=-8 is not a solution.

x=8

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Example Question #113 : Equations

For what value of is $2^{x-4}=16$ ?

Possible Answers:

x=7

x=6

x=5

x=9

x=8

Correct answer:

x=8

Explanation:

$2^{x-4}=16\Rightarrow 2^{x-4}=2^4$

Since the bases are the same, the two expressions can only be equal if the powers are the same:

$x-4=4\Rightarrow x-4+4=4+4\Rightarrow x=8$

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Example Question #112 : How To Find The Solution To An Equation

Solve for :

$10^{3-x}=(5\times 2)^4$

Possible Answers:

x=2

x=1

x=-2

x=-1

x=3

Correct answer:

x=-1

Explanation:

$10^{3-x}=(5\times 2)^4\Rightarrow 10^{3-x}=(10)^4$

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

$3-x=4\Rightarrow x=3-4=-1$

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Example Question #115 : Equations

Solve for :

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

Possible Answers:

x=1

x=-4

x=5

x=3

x=4

Correct answer:

x=4

Explanation:

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

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

$2x-3=3x-7\Rightarrow 2x-3+3=3x-7+3$

$\Rightarrow 2x=3x-4\Rightarrow 3x-2x=4\Rightarrow x=4$

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Example Question #116 : Equations

Solve for :

$3^{2a+1}=27^a$

Possible Answers:

a=0

a=-1

a=2

a=3

a=1

Correct answer:

a=1

Explanation:

$3^{2a+1}=27^a\Rightarrow 3^{2a+1}=(3^3)^a\Rightarrow 3^{2a+1}=3^{3a}$

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

$2a+1=3a\Rightarrow 3a-2a=1\Rightarrow a=1$

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Example Question #117 : Equations

Solve the following equation for :

$5^{4x-7}=125^{3x}$

Possible Answers:

x=1.8

x=1.5

x=1

x=1.4

x=-1.4

Correct answer:

x=-1.4

Explanation:

$5^{4x-7}=125^{3x}\Rightarrow 5^{4x-7}=\left [5^3 \right ]^{3x}\Rightarrow 5^{4x-7}=5^{3\times 3x}$

$\Rightarrow 5^{4x-7}=5^{9x}$

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

$4x-7=9x\Rightarrow 9x-4x=-7\Rightarrow 5x=-7\Rightarrow x=\frac{-7}{5}=-1.4$

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Example Question #118 : Equations

Solve the following equation for :

$\left [ 3^x \right ]^{(4x-5)}=81^{x^2+2x-1}$

Possible Answers:

$x=-\frac{4}{13}$

$x=\frac{13}{4}$

x=13

$x=\frac{4}{13}$

x=4

Correct answer:

$x=\frac{4}{13}$

Explanation:

$\left [ 3^x \right ]^{(4x-5)}=81^{x^2+2x-1}\Rightarrow 3^{x(4x-5)}=\left [3^4 \right ]^{x^2+2x-1}$

$\Rightarrow 3^{4x^2-5x}=3^{4(x^2+2x-1)}$

$\Rightarrow 3^{4x^2-5x}=3^{4x^2+8x-4}$

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

4x^2-5x=4x^2+8x-4

$\Rightarrow 4x^2-4x^2-5x-8x=-4$

$\Rightarrow -13x=-4$

$\Rightarrow x=\frac{-4}{-13}=\frac{4}{13}$

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Example Question #119 : Equations

Solve for :

$4^{2x-1}=8^{4x+2}$

Possible Answers:

x=-1

x=0

x=1

x=2

x=-2

Correct answer:

x=-1

Explanation:

We know that 4=2^2 and 8=2^3 .

$4^{2x-1}=8^{4x+2}\Rightarrow \left [ 2^2 \right ]^{2x-1}=\left [ 2^3 \right ]^{4x+2}$

$\Rightarrow 2^{2(2x-1)}=2^{3(4x+2)}$

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

$2(2x-1)=3(4x+2)\Rightarrow 4x-2=12x+6$

$\Rightarrow 4x-12x=6+2$

$\Rightarrow -8x=8$

$\Rightarrow x=\frac{8}{-8}=-1$

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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 #111 : Algebraic Concepts

Example Question #111 : Equations

Example Question #112 : Equations

Example Question #113 : Equations

Example Question #112 : How To Find The Solution To An Equation

Example Question #115 : Equations

Example Question #116 : Equations

Example Question #117 : Equations

Example Question #118 : Equations

Example Question #119 : Equations

All ISEE Upper Level Math Resources

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