Finite Math Examples

$0.5 e^{- 0.5 x} = 10^{- 5}$

Step 1

Divide each term in $0.5 e^{- 0.5 x} = 10^{- 5}$ by $0.5$ and simplify.

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Step 1.1

Divide each term in $0.5 e^{- 0.5 x} = 10^{- 5}$ by $0.5$ .

$\frac{0.5 e^{- 0.5 x}}{0.5} = \frac{10^{- 5}}{0.5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $0.5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{0.5 e^{- 0.5 x}}{0.5} = \frac{10^{- 5}}{0.5}$

Step 1.2.1.2

Divide $e^{- 0.5 x}$ by $1$ .

$e^{- 0.5 x} = \frac{10^{- 5}}{0.5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move $10^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{- 0.5 x} = \frac{1}{0.5 \cdot 10^{5}}$

Step 1.3.2

Divide using scientific notation.

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Step 1.3.2.1

Group coefficients together and exponents together to divide numbers in scientific notation.

$e^{- 0.5 x} = (\frac{1}{0.5}) (\frac{1}{10^{5}})$

Step 1.3.2.2

Divide $1$ by $0.5$ .

$e^{- 0.5 x} = 2 \frac{1}{10^{5}}$

Step 1.3.2.3

Move $10^{5}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$e^{- 0.5 x} = 2 \cdot 10^{- 5}$

Step 2

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

$\ln (e^{- 0.5 x}) = \ln (2 \cdot 10^{- 5})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{- 0.5 x})$ by moving $- 0.5 x$ outside the logarithm.

$- 0.5 x \ln (e) = \ln (2 \cdot 10^{- 5})$

Step 3.2

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

$- 0.5 x \cdot 1 = \ln (2 \cdot 10^{- 5})$

Step 3.3

Multiply $- 0.5$ by $1$ .

$- 0.5 x = \ln (2 \cdot 10^{- 5})$

Step 4

Divide each term in $- 0.5 x = \ln (2 \cdot 10^{- 5})$ by $- 0.5$ and simplify.

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Step 4.1

Divide each term in $- 0.5 x = \ln (2 \cdot 10^{- 5})$ by $- 0.5$ .

$\frac{- 0.5 x}{- 0.5} = \frac{\ln (2 \cdot 10^{- 5})}{- 0.5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.5 x}{- 0.5} = \frac{\ln (2 \cdot 10^{- 5})}{- 0.5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2 \cdot 10^{- 5})}{- 0.5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (2 \cdot 10^{- 5})}{0.5}$

Step 4.3.2

Multiply by $1$ .

$x = - \frac{1 \ln (2 \cdot 10^{- 5})}{0.5}$

Step 4.3.3

Factor $0.5$ out of $0.5$ .

$x = - \frac{1 \ln (2 \cdot 10^{- 5})}{0.5 (1)}$

Step 4.3.4

Separate fractions.

$x = - (\frac{1}{0.5} \cdot \frac{\ln (2 \cdot 10^{- 5})}{1})$

Step 4.3.5

Divide $1$ by $0.5$ .

$x = - (2 \frac{\ln (2 \cdot 10^{- 5})}{1})$

Step 4.3.6

Divide $\ln (2 \cdot 10^{- 5})$ by $1$ .

$x = - (2 \ln (2 \cdot 10^{- 5}))$

Step 4.3.7

Multiply $2$ by $- 1$ .

$x = - 2 \ln (2 \cdot 10^{- 5})$

Step 5

The result can be shown in multiple forms.

Scientific Notation:

$x = - 2 \ln (2 \cdot 10^{- 5})$

Expanded Form:

$x = 21.63955656$