Solve for x -2*8^(4-x)=-50

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

Step 1

Simplify.

Factor out negative.

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $3$ by $4$ .

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

Multiply $- 1$ by $3$ .

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

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

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

Add $1$ and $12$ .

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

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $2^{13 - 3 x}$ and $- 2$ .

Rewrite $- 2$ as $- 1 (2)$ .

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

Factor $2$ out of $- 2^{13 - 3 x}$ .

$\frac{2 (- 2^{- 3 x + 12})}{- 1 (2)} = \frac{- 50}{- 2}$

Move the negative one from the denominator of $\frac{- 2^{- 3 x + 12}}{- 1}$ .

$- 1 \cdot (- 2^{- 3 x + 12}) = \frac{- 50}{- 2}$

Rewrite $- 1 \cdot (- 2^{- 3 x + 12})$ as $- (- 2^{- 3 x + 12})$ .

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

Multiply $- (- 2^{- 3 x + 12})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $2^{- 3 x + 12}$ by $1$ .

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

Simplify the right side.

Divide $- 50$ by $- 2$ .

$2^{- 3 x + 12} = 25$

Step 3

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

$\ln (2^{- 3 x + 12}) = \ln (25)$

Step 4

Expand $\ln (2^{- 3 x + 12})$ by moving $- 3 x + 12$ outside the logarithm.

$(- 3 x + 12) \ln (2) = \ln (25)$

Step 5

Simplify the left side.

Apply the distributive property.

$- 3 x \ln (2) + 12 \ln (2) = \ln (25)$

Step 6

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

$- 3 x \ln (2) + 12 \ln (2) - \ln (25) = 0$

Step 7

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

Subtract $12 \ln (2)$ from both sides of the equation.

$- 3 x \ln (2) - \ln (25) = - 12 \ln (2)$

Add $\ln (25)$ to both sides of the equation.

$- 3 x \ln (2) = - 12 \ln (2) + \ln (25)$

Step 8

Divide each term in $- 3 x \ln (2) = - 12 \ln (2) + \ln (25)$ by $- 3 \ln (2)$ and simplify.

Divide each term in $- 3 x \ln (2) = - 12 \ln (2) + \ln (25)$ by $- 3 \ln (2)$ .

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 12 \ln (2)}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x \ln (2)}{- 3 \ln (2)} = \frac{- 12 \ln (2)}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 12 \ln (2)}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

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

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 12 \ln (2)}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{- 12 \ln (2)}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 12$ and $- 3$ .

Factor $- 3$ out of $- 12 \ln (2)$ .

$x = \frac{- 3 (4 \ln (2))}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Cancel the common factors.

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

$x = \frac{- 3 (4 \ln (2))}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Cancel the common factor.

$x = \frac{- 3 (4 \ln (2))}{- 3 \ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Rewrite the expression.

$x = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

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

Cancel the common factor.

$x = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (25)}{- 3 \ln (2)}$

Divide $4$ by $1$ .

$x = 4 + \frac{\ln (25)}{- 3 \ln (2)}$

Move the negative in front of the fraction.

$x = 4 - \frac{\ln (25)}{3 \ln (2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 4 - \frac{\ln (25)}{3 \ln (2)}$

Decimal Form:

$x = 2.45204793 \dots$

Solve for x -2*8^(4-x)=-50

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