Trigonometry Examples

$4.5 = 2100000 (1 - e^{- 0.03 x})$

Step 1

Rewrite the equation as $2100000 (1 - e^{- 0.03 x}) = 4.5$ .

$2100000 (1 - e^{- 0.03 x}) = 4.5$

Step 2

Divide each term in $2100000 (1 - e^{- 0.03 x}) = 4.5$ by $2100000$ and simplify.

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Step 2.1

Divide each term in $2100000 (1 - e^{- 0.03 x}) = 4.5$ by $2100000$ .

$\frac{2100000 (1 - e^{- 0.03 x})}{2100000} = \frac{4.5}{2100000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2100000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2100000 (1 - e^{- 0.03 x})}{2100000} = \frac{4.5}{2100000}$

Step 2.2.1.2

Divide $1 - e^{- 0.03 x}$ by $1$ .

$1 - e^{- 0.03 x} = \frac{4.5}{2100000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $4.5$ by $2100000$ .

$1 - e^{- 0.03 x} = 0.00000214$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $1$ from both sides of the equation.

$- e^{- 0.03 x} = 0.00000214 - 1$

Step 3.2

Subtract $1$ from $0.00000214$ .

$- e^{- 0.03 x} = - 0.99999785$

Step 4

Divide each term in $- e^{- 0.03 x} = - 0.99999785$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- e^{- 0.03 x} = - 0.99999785$ by $- 1$ .

$\frac{- e^{- 0.03 x}}{- 1} = \frac{- 0.99999785}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- 0.03 x}}{1} = \frac{- 0.99999785}{- 1}$

Step 4.2.2

Divide $e^{- 0.03 x}$ by $1$ .

$e^{- 0.03 x} = \frac{- 0.99999785}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 0.99999785$ by $- 1$ .

$e^{- 0.03 x} = 0.99999785$

Step 5

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

$\ln (e^{- 0.03 x}) = \ln (0.99999785)$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{- 0.03 x})$ by moving $- 0.03 x$ outside the logarithm.

$- 0.03 x \ln (e) = \ln (0.99999785)$

Step 6.2

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

$- 0.03 x \cdot 1 = \ln (0.99999785)$

Step 6.3

Multiply $- 0.03$ by $1$ .

$- 0.03 x = \ln (0.99999785)$

Step 7

Divide each term in $- 0.03 x = \ln (0.99999785)$ by $- 0.03$ and simplify.

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Step 7.1

Divide each term in $- 0.03 x = \ln (0.99999785)$ by $- 0.03$ .

$\frac{- 0.03 x}{- 0.03} = \frac{\ln (0.99999785)}{- 0.03}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 0.03$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 0.03 x}{- 0.03} = \frac{\ln (0.99999785)}{- 0.03}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (0.99999785)}{- 0.03}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.99999785)}{0.03}$

Step 7.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (0.99999785)}{0.03}$

Step 7.3.3

Log base $2.71828182$ of $0.99999785$ is approximately $- 0.00000214$ .

$x = - \frac{- 0.00000214}{0.03}$

Step 7.3.4

Divide $- 0.00000214$ by $0.03$ .

$x = - - 0.00007142$

Step 7.3.5

Multiply $- 1$ by $- 0.00007142$ .

$x = 0.00007142$