Precalculus Examples

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

Step 1

Subtract $\frac{1}{4}$ from both sides of the equation.

$2^{7 - 3 x} - \frac{1}{4} = 0$

Step 2

Simplify $2^{7 - 3 x} - \frac{1}{4}$ .

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Step 2.1

To write $2^{7 - 3 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2^{7 - 3 x} \cdot \frac{4}{4} - \frac{1}{4} = 0$

Step 2.2

Combine $2^{7 - 3 x}$ and $\frac{4}{4}$ .

$\frac{2^{7 - 3 x} \cdot 4}{4} - \frac{1}{4} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{2^{7 - 3 x} \cdot 4 - 1}{4} = 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{7 - 3 x} \cdot 2^{2} - 1}{4} = 0$

Step 2.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{7 - 3 x + 2} - 1}{4} = 0$

Step 2.4.3

Add $7$ and $2$ .

$\frac{2^{- 3 x + 9} - 1}{4} = 0$

Step 2.4.4

Rewrite $2^{- 3 x + 9}$ as ${(2^{- x + 3})}^{3}$ .

$\frac{{(2^{- x + 3})}^{3} - 1}{4} = 0$

Step 2.4.5

Rewrite $1$ as $1^{3}$ .

$\frac{{(2^{- x + 3})}^{3} - 1^{3}}{4} = 0$

Step 2.4.6

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2^{- x + 3}$ and $b = 1$ .

$\frac{(2^{- x + 3} - 1) ({(2^{- x + 3})}^{2} + 2^{- x + 3} \cdot 1 + 1^{2})}{4} = 0$

Step 2.4.7

Simplify.

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Step 2.4.7.1

Multiply the exponents in ${(2^{- x + 3})}^{2}$ .

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Step 2.4.7.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(2^{- x + 3} - 1) (2^{(- x + 3) \cdot 2} + 2^{- x + 3} \cdot 1 + 1^{2})}{4} = 0$

Step 2.4.7.1.2

Apply the distributive property.

$\frac{(2^{- x + 3} - 1) (2^{- x \cdot 2 + 3 \cdot 2} + 2^{- x + 3} \cdot 1 + 1^{2})}{4} = 0$

Step 2.4.7.1.3

Multiply $2$ by $- 1$ .

$\frac{(2^{- x + 3} - 1) (2^{- 2 x + 3 \cdot 2} + 2^{- x + 3} \cdot 1 + 1^{2})}{4} = 0$

Step 2.4.7.1.4

Multiply $3$ by $2$ .

$\frac{(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} \cdot 1 + 1^{2})}{4} = 0$

Step 2.4.7.2

Multiply $2^{- x + 3}$ by $1$ .

$\frac{(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} + 1^{2})}{4} = 0$

Step 2.4.7.3

One to any power is one.

$\frac{(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} + 1)}{4} = 0$

Step 3

Set the numerator equal to zero.

$(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} + 1) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} + 1)$ .

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Step 4.1.1

Expand $(2^{- x + 3} - 1) (2^{- 2 x + 6} + 2^{- x + 3} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} + 2^{- x + 3} \cdot 1 - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 2^{- x + 3} - 1 \cdot 1 = 0$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Combine the opposite terms in $2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} + 2^{- x + 3} \cdot 1 - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 2^{- x + 3} - 1 \cdot 1$ .

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Step 4.1.2.1.1

Reorder the factors in the terms $2^{- x + 3} \cdot 1$ and $- 1 \cdot 2^{- x + 3}$ .

$2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} + 1 \cdot 2^{- x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 2^{- x + 3} - 1 \cdot 1 = 0$

Step 4.1.2.1.2

Subtract $1 \cdot 2^{- x + 3}$ from $1 \cdot 2^{- x + 3}$ .

$2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} + 0 - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.1.3

Add $2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3}$ and $0$ .

$2^{- x + 3} \cdot 2^{- 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

Multiply $2^{- x + 3}$ by $2^{- 2 x + 6}$ by adding the exponents.

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Step 4.1.2.2.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{- x + 3 - 2 x + 6} + 2^{- x + 3} \cdot 2^{- x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.1.2

Subtract $2 x$ from $- x$ .

$2^{- 3 x + 3 + 6} + 2^{- x + 3} \cdot 2^{- x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.1.3

Add $3$ and $6$ .

$2^{- 3 x + 9} + 2^{- x + 3} \cdot 2^{- x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.2

Multiply $2^{- x + 3}$ by $2^{- x + 3}$ by adding the exponents.

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Step 4.1.2.2.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{- 3 x + 9} + 2^{- x + 3 - x + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.2.2

Subtract $x$ from $- x$ .

$2^{- 3 x + 9} + 2^{- 2 x + 3 + 3} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.2.3

Add $3$ and $3$ .

$2^{- 3 x + 9} + 2^{- 2 x + 6} - 1 \cdot 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.3

Rewrite $- 1 \cdot 2^{- 2 x + 6}$ as $- 2^{- 2 x + 6}$ .

$2^{- 3 x + 9} + 2^{- 2 x + 6} - 2^{- 2 x + 6} - 1 \cdot 1 = 0$

Step 4.1.2.2.4

Multiply $- 1$ by $1$ .

$2^{- 3 x + 9} + 2^{- 2 x + 6} - 2^{- 2 x + 6} - 1 = 0$

Step 4.1.2.3

Combine the opposite terms in $2^{- 3 x + 9} + 2^{- 2 x + 6} - 2^{- 2 x + 6} - 1$ .

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Step 4.1.2.3.1

Subtract $2^{- 2 x + 6}$ from $2^{- 2 x + 6}$ .

$2^{- 3 x + 9} + 0 - 1 = 0$

Step 4.1.2.3.2

Add $2^{- 3 x + 9}$ and $0$ .

$2^{- 3 x + 9} - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$2^{- 3 x + 9} = 1$

Step 4.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{- 3 x + 9}) = \ln (1)$

Step 4.4

Expand $\ln (2^{- 3 x + 9})$ by moving $- 3 x + 9$ outside the logarithm.

$(- 3 x + 9) \ln (2) = \ln (1)$

Step 4.5

Simplify the left side.

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Step 4.5.1

Apply the distributive property.

$- 3 x \ln (2) + 9 \ln (2) = \ln (1)$

Step 4.6

Simplify the right side.

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Step 4.6.1

The natural logarithm of $1$ is $0$ .

$- 3 x \ln (2) + 9 \ln (2) = 0$

Step 4.7

Subtract $9 \ln (2)$ from both sides of the equation.

$- 3 x \ln (2) = - 9 \ln (2)$

Step 4.8

Divide each term in $- 3 x \ln (2) = - 9 \ln (2)$ by $- 3 \ln (2)$ and simplify.

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Step 4.8.1

Divide each term in $- 3 x \ln (2) = - 9 \ln (2)$ by $- 3 \ln (2)$ .

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 9 \ln (2)}{- 3 \ln (2)}$

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $- 3$ .

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Step 4.8.2.1.1

Cancel the common factor.

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 9 \ln (2)}{- 3 \ln (2)}$

Step 4.8.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 9 \ln (2)}{- 3 \ln (2)}$

Step 4.8.2.2

Cancel the common factor of $\ln (2)$ .

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Step 4.8.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 9 \ln (2)}{- 3 \ln (2)}$

Step 4.8.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 9 \ln (2)}{- 3 \ln (2)}$

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Cancel the common factor of $- 9$ and $- 3$ .

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Step 4.8.3.1.1

Factor $- 3$ out of $- 9 \ln (2)$ .

$x = \frac{- 3 (3 \ln (2))}{- 3 \ln (2)}$

Step 4.8.3.1.2

Cancel the common factors.

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Step 4.8.3.1.2.1

Factor $- 3$ out of $- 3 \ln (2)$ .

$x = \frac{- 3 (3 \ln (2))}{- 3 \ln (2)}$

Step 4.8.3.1.2.2

Cancel the common factor.

$x = \frac{- 3 (3 \ln (2))}{- 3 \ln (2)}$

Step 4.8.3.1.2.3

Rewrite the expression.

$x = \frac{3 \ln (2)}{\ln (2)}$

Step 4.8.3.2

Cancel the common factor of $\ln (2)$ .

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Step 4.8.3.2.1

Cancel the common factor.

$x = \frac{3 \ln (2)}{\ln (2)}$

Step 4.8.3.2.2

Divide $3$ by $1$ .

$x = 3$