Precalculus Examples

$8^{- 2 - x} = 431$

Step 1

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

$\ln (8^{- 2 - x}) = \ln (431)$

Step 2

Expand $\ln (8^{- 2 - x})$ by moving $- 2 - x$ outside the logarithm.

$(- 2 - x) \ln (8) = \ln (431)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$- 2 \ln (8) - x \ln (8) = \ln (431)$

Step 4

Reorder $- 2 \ln (8)$ and $- x \ln (8)$ .

$- x \ln (8) - 2 \ln (8) = \ln (431)$

Step 5

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

$- x \ln (8) - 2 \ln (8) - \ln (431) = 0$

Step 6

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

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Step 6.1

Add $2 \ln (8)$ to both sides of the equation.

$- x \ln (8) - \ln (431) = 2 \ln (8)$

Step 6.2

Add $\ln (431)$ to both sides of the equation.

$- x \ln (8) = 2 \ln (8) + \ln (431)$

Step 7

Divide each term in $- x \ln (8) = 2 \ln (8) + \ln (431)$ by $- \ln (8)$ and simplify.

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Step 7.1

Divide each term in $- x \ln (8) = 2 \ln (8) + \ln (431)$ by $- \ln (8)$ .

$\frac{- x \ln (8)}{- \ln (8)} = \frac{2 \ln (8)}{- \ln (8)} + \frac{\ln (431)}{- \ln (8)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (8)}{\ln (8)} = \frac{2 \ln (8)}{- \ln (8)} + \frac{\ln (431)}{- \ln (8)}$

Step 7.2.2

Cancel the common factor of $\ln (8)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{x \ln (8)}{\ln (8)} = \frac{2 \ln (8)}{- \ln (8)} + \frac{\ln (431)}{- \ln (8)}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (8)}{- \ln (8)} + \frac{\ln (431)}{- \ln (8)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $\ln (8)$ .

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Step 7.3.1.1.1

Cancel the common factor.

$x = \frac{2 \ln (8)}{- \ln (8)} + \frac{\ln (431)}{- \ln (8)}$

Step 7.3.1.1.2

Rewrite the expression.

$x = \frac{2}{- 1} + \frac{\ln (431)}{- \ln (8)}$

Step 7.3.1.1.3

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$x = - 1 \cdot 2 + \frac{\ln (431)}{- \ln (8)}$

Step 7.3.1.2

Multiply $- 1$ by $2$ .

$x = - 2 + \frac{\ln (431)}{- \ln (8)}$

Step 7.3.1.3

Move the negative in front of the fraction.

$x = - 2 - \frac{\ln (431)}{\ln (8)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 2 - \frac{\ln (431)}{\ln (8)}$

Decimal Form:

$x = - 4.91718135 \dots$