Algebra Examples

$π^{1 - x} = e^{x}$

Step 1

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

$\ln (π^{1 - x}) = \ln (e^{x})$

Step 2

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

$(1 - x) \ln (π) = \ln (e^{x})$

Step 3

Expand the right side.

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Step 3.1

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

$(1 - x) \ln (π) = x \ln (e)$

Step 3.2

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

$(1 - x) \ln (π) = x \cdot 1$

Step 3.3

Multiply $x$ by $1$ .

$(1 - x) \ln (π) = x$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(1 - x) \ln (π)$ .

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Step 4.1.1

Apply the distributive property.

$1 \ln (π) - x \ln (π) = x$

Step 4.1.2

Multiply $\ln (π)$ by $1$ .

$\ln (π) - x \ln (π) = x$

Step 5

Reorder $\ln (π)$ and $- x \ln (π)$ .

$- x \ln (π) + \ln (π) = x$

Step 6

Subtract $x$ from both sides of the equation.

$- x \ln (π) + \ln (π) - x = 0$

Step 7

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

$- x \ln (π) - x = - \ln (π)$

Step 8

Factor $x$ out of $- x \ln (π) - x$ .

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Step 8.1

Factor $x$ out of $- x \ln (π)$ .

$x (- 1 \ln (π)) - x = - \ln (π)$

Step 8.2

Factor $x$ out of $- x$ .

$x (- 1 \ln (π)) + x \cdot - 1 = - \ln (π)$

Step 8.3

Factor $x$ out of $x (- 1 \ln (π)) + x \cdot - 1$ .

$x (- 1 \ln (π) - 1) = - \ln (π)$

Step 9

Rewrite $- 1 \ln (π)$ as $- \ln (π)$ .

$x (- \ln (π) - 1) = - \ln (π)$

Step 10

Divide each term in $x (- \ln (π) - 1) = - \ln (π)$ by $- \ln (π) - 1$ and simplify.

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Step 10.1

Divide each term in $x (- \ln (π) - 1) = - \ln (π)$ by $- \ln (π) - 1$ .

$\frac{x (- \ln (π) - 1)}{- \ln (π) - 1} = \frac{- \ln (π)}{- \ln (π) - 1}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $- \ln (π) - 1$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{x (- \ln (π) - 1)}{- \ln (π) - 1} = \frac{- \ln (π)}{- \ln (π) - 1}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (π)}{- \ln (π) - 1}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (π)}{- \ln (π) - 1}$

Step 10.3.2

Factor $- 1$ out of $- \ln (π)$ .

$x = - \frac{\ln (π)}{- \ln (π) - 1}$

Step 10.3.3

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

$x = - \frac{\ln (π)}{- \ln (π) - 1 (1)}$

Step 10.3.4

Factor $- 1$ out of $- \ln (π) - 1 (1)$ .

$x = - \frac{\ln (π)}{- (\ln (π) + 1)}$

Step 10.3.5

Simplify the expression.

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Step 10.3.5.1

Rewrite $- (\ln (π) + 1)$ as $- 1 (\ln (π) + 1)$ .

$x = - \frac{\ln (π)}{- 1 (\ln (π) + 1)}$

Step 10.3.5.2

Move the negative in front of the fraction.

$x = - - \frac{\ln (π)}{\ln (π) + 1}$

Step 10.3.5.3

Multiply $- 1$ by $- 1$ .

$x = 1 \frac{\ln (π)}{\ln (π) + 1}$

Step 10.3.5.4

Multiply $\frac{\ln (π)}{\ln (π) + 1}$ by $1$ .

$x = \frac{\ln (π)}{\ln (π) + 1}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (π)}{\ln (π) + 1}$

Decimal Form:

$x = 0.53374081 \dots$