Finite Math Examples

$0.11 x = x (2^{- x})$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x (2^{- x}) = 0.11 x$

Step 2

Subtract $0.11 x$ from both sides of the equation.

$x (2^{- x}) - 0.11 x = 0$

Step 3

Factor $x$ out of $x (2^{- x}) - 0.11 x$ .

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Step 3.1

Factor $x$ out of $- 0.11 x$ .

$x (2^{- x}) + x \cdot - 0.11 = 0$

Step 3.2

Factor $x$ out of $x (2^{- x}) + x \cdot - 0.11$ .

$x (2^{- x} - 0.11) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2^{- x} - 0.11 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $2^{- x} - 0.11$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2^{- x} - 0.11$ equal to $0$ .

$2^{- x} - 0.11 = 0$

Step 6.2

Solve $2^{- x} - 0.11 = 0$ for $x$ .

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Step 6.2.1

Add $0.11$ to both sides of the equation.

$2^{- x} = 0.11$

Step 6.2.2

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

$\ln (2^{- x}) = \ln (0.11)$

Step 6.2.3

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

$- x \ln (2) = \ln (0.11)$

Step 6.2.4

Divide each term in $- x \ln (2) = \ln (0.11)$ by $- \ln (2)$ and simplify.

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Step 6.2.4.1

Divide each term in $- x \ln (2) = \ln (0.11)$ by $- \ln (2)$ .

$\frac{- x \ln (2)}{- \ln (2)} = \frac{\ln (0.11)}{- \ln (2)}$

Step 6.2.4.2

Simplify the left side.

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Step 6.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (0.11)}{- \ln (2)}$

Step 6.2.4.2.2

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

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Step 6.2.4.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (0.11)}{- \ln (2)}$

Step 6.2.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (0.11)}{- \ln (2)}$

Step 6.2.4.3

Simplify the right side.

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Step 6.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.11)}{\ln (2)}$

Step 7

The final solution is all the values that make $x (2^{- x} - 0.11) = 0$ true.

$x = 0, - \frac{\ln (0.11)}{\ln (2)}$

Step 8