Pre-Algebra Examples

$2^{4} - {2.2}^{1 - x} = 0$

Step 1

Raise $2$ to the power of $4$ .

$16 - {2.2}^{1 - x} = 0$

Step 2

Subtract $16$ from both sides of the equation.

$- {2.2}^{1 - x} = - 16$

Step 3

Divide each term in $- {2.2}^{1 - x} = - 16$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- {2.2}^{1 - x} = - 16$ by $- 1$ .

$\frac{- {2.2}^{1 - x}}{- 1} = \frac{- 16}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{{2.2}^{1 - x}}{1} = \frac{- 16}{- 1}$

Step 3.2.2

Divide ${2.2}^{1 - x}$ by $1$ .

${2.2}^{1 - x} = \frac{- 16}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 16$ by $- 1$ .

${2.2}^{1 - x} = 16$

Step 4

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

$\ln ({2.2}^{1 - x}) = \ln (16)$

Step 5

Expand $\ln ({2.2}^{1 - x})$ by moving $1 - x$ outside the logarithm.

$(1 - x) \ln (2.2) = \ln (16)$

Step 6

Simplify the left side.

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Step 6.1

Simplify $(1 - x) \ln (2.2)$ .

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Step 6.1.1

Apply the distributive property.

$1 \ln (2.2) - x \ln (2.2) = \ln (16)$

Step 6.1.2

Multiply $\ln (2.2)$ by $1$ .

$\ln (2.2) - x \ln (2.2) = \ln (16)$

Step 7

Reorder $\ln (2.2)$ and $- x \ln (2.2)$ .

$- x \ln (2.2) + \ln (2.2) = \ln (16)$

Step 8

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

$- x \ln (2.2) + \ln (2.2) - \ln (16) = 0$

Step 9

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- x \ln (2.2) + \ln (\frac{2.2}{16}) = 0$

Step 10

Divide $2.2$ by $16$ .

$- x \ln (2.2) + \ln (0.1375) = 0$

Step 11

Subtract $\ln (0.1375)$ from both sides of the equation.

$- x \ln (2.2) = - \ln (0.1375)$

Step 12

Divide each term in $- x \ln (2.2) = - \ln (0.1375)$ by $- \ln (2.2)$ and simplify.

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Step 12.1

Divide each term in $- x \ln (2.2) = - \ln (0.1375)$ by $- \ln (2.2)$ .

$\frac{- x \ln (2.2)}{- \ln (2.2)} = \frac{- \ln (0.1375)}{- \ln (2.2)}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (2.2)}{\ln (2.2)} = \frac{- \ln (0.1375)}{- \ln (2.2)}$

Step 12.2.2

Cancel the common factor of $\ln (2.2)$ .

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Step 12.2.2.1

Cancel the common factor.

$\frac{x \ln (2.2)}{\ln (2.2)} = \frac{- \ln (0.1375)}{- \ln (2.2)}$

Step 12.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.1375)}{- \ln (2.2)}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Dividing two negative values results in a positive value.

$x = \frac{\ln (0.1375)}{\ln (2.2)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (0.1375)}{\ln (2.2)}$

Decimal Form:

$x = - 2.51647262 \dots$