Algebra Examples

$- 2 \cdot 8^{4 - x} = - 50$

Step 1

Simplify.

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Step 1.1

Factor out negative.

$- (2 \cdot 8^{4 - x}) = - 50$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$- (2 \cdot {(2^{3})}^{4 - x}) = - 50$

Step 1.3

Multiply the exponents in ${(2^{3})}^{4 - x}$ .

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Step 1.3.1

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

$- (2 \cdot 2^{3 (4 - x)}) = - 50$

Step 1.3.2

Apply the distributive property.

$- (2 \cdot 2^{3 \cdot 4 + 3 (- x)}) = - 50$

Step 1.3.3

Multiply $3$ by $4$ .

$- (2 \cdot 2^{12 + 3 (- x)}) = - 50$

Step 1.3.4

Multiply $- 1$ by $3$ .

$- (2 \cdot 2^{12 - 3 x}) = - 50$

Step 1.4

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

$- 2^{1 + 12 - 3 x} = - 50$

Step 1.5

Add $1$ and $12$ .

$- 2^{13 - 3 x} = - 50$

Step 2

Divide each term in $- 2^{13 - 3 x} = - 50$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- 2^{13 - 3 x} = - 50$ by $- 1$ .

$\frac{- 2^{13 - 3 x}}{- 1} = \frac{- 50}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{2^{13 - 3 x}}{1} = \frac{- 50}{- 1}$

Step 2.2.2

Divide $2^{13 - 3 x}$ by $1$ .

$2^{13 - 3 x} = \frac{- 50}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 50$ by $- 1$ .

$2^{13 - 3 x} = 50$

Step 3

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

$\ln (2^{13 - 3 x}) = \ln (50)$

Step 4

Expand $\ln (2^{13 - 3 x})$ by moving $13 - 3 x$ outside the logarithm.

$(13 - 3 x) \ln (2) = \ln (50)$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$13 \ln (2) - 3 x \ln (2) = \ln (50)$

Step 6

Reorder $13 \ln (2)$ and $- 3 x \ln (2)$ .

$- 3 x \ln (2) + 13 \ln (2) = \ln (50)$

Step 7

Move all the terms containing a logarithm to the left side of the equation.

$- 3 x \ln (2) + 13 \ln (2) - \ln (50) = 0$

Step 8

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1

Subtract $13 \ln (2)$ from both sides of the equation.

$- 3 x \ln (2) - \ln (50) = - 13 \ln (2)$

Step 8.2

Add $\ln (50)$ to both sides of the equation.

$- 3 x \ln (2) = - 13 \ln (2) + \ln (50)$

Step 9

Divide each term in $- 3 x \ln (2) = - 13 \ln (2) + \ln (50)$ by $- 3 \ln (2)$ and simplify.

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Step 9.1

Divide each term in $- 3 x \ln (2) = - 13 \ln (2) + \ln (50)$ by $- 3 \ln (2)$ .

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 3$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.2.2

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

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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Step 9.3.1.1.1

Cancel the common factor.

$x = \frac{- 13 \ln (2)}{- 3 \ln (2)} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.3.1.1.2

Rewrite the expression.

$x = \frac{- 13}{- 3} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{13}{3} + \frac{\ln (50)}{- 3 \ln (2)}$

Step 9.3.1.3

Move the negative in front of the fraction.

$x = \frac{13}{3} - \frac{\ln (50)}{3 \ln (2)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{3} - \frac{\ln (50)}{3 \ln (2)}$

Decimal Form:

$x = 2.45204793 \dots$