Finite Math Examples

$1.75 = 100 e^{- \frac{x}{2}}$

Step 1

Rewrite the equation as $100 e^{- \frac{x}{2}} = 1.75$ .

$100 e^{- \frac{x}{2}} = 1.75$

Step 2

Divide each term in $100 e^{- \frac{x}{2}} = 1.75$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 e^{- \frac{x}{2}} = 1.75$ by $100$ .

$\frac{100 e^{- \frac{x}{2}}}{100} = \frac{1.75}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 e^{- \frac{x}{2}}}{100} = \frac{1.75}{100}$

Step 2.2.1.2

Divide $e^{- \frac{x}{2}}$ by $1$ .

$e^{- \frac{x}{2}} = \frac{1.75}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $1.75$ by $100$ .

$e^{- \frac{x}{2}} = 0.0175$

Step 3

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

$\ln (e^{- \frac{x}{2}}) = \ln (0.0175)$

Step 4

Expand the left side.

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Step 4.1

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

$- \frac{x}{2} \ln (e) = \ln (0.0175)$

Step 4.2

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

$- \frac{x}{2} \cdot 1 = \ln (0.0175)$

Step 4.3

Multiply $- 1$ by $1$ .

$- \frac{x}{2} = \ln (0.0175)$

Step 5

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{x}{2}) = - 2 \ln (0.0175)$

Step 6

Simplify the left side.

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Step 6.1

Simplify $- 2 (- \frac{x}{2})$ .

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$- 2 \frac{- x}{2} = - 2 \ln (0.0175)$

Step 6.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- x}{2} = - 2 \ln (0.0175)$

Step 6.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- x}{2} = - 2 \ln (0.0175)$

Step 6.1.1.4

Rewrite the expression.

$- - x = - 2 \ln (0.0175)$

Step 6.1.2

Multiply.

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 2 \ln (0.0175)$

Step 6.1.2.2

Multiply $x$ by $1$ .

$x = - 2 \ln (0.0175)$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 2 \ln (0.0175)$

Decimal Form:

$x = 8.09110879 \dots$