Trigonometry Examples

$1.6 e^{- 1.3 x} = 1$

Step 1

Divide each term in $1.6 e^{- 1.3 x} = 1$ by $1.6$ and simplify.

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Step 1.1

Divide each term in $1.6 e^{- 1.3 x} = 1$ by $1.6$ .

$\frac{1.6 e^{- 1.3 x}}{1.6} = \frac{1}{1.6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $1.6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1.6 e^{- 1.3 x}}{1.6} = \frac{1}{1.6}$

Step 1.2.1.2

Divide $e^{- 1.3 x}$ by $1$ .

$e^{- 1.3 x} = \frac{1}{1.6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $1$ by $1.6$ .

$e^{- 1.3 x} = 0.625$

Step 2

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

$\ln (e^{- 1.3 x}) = \ln (0.625)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{- 1.3 x})$ by moving $- 1.3 x$ outside the logarithm.

$- 1.3 x \ln (e) = \ln (0.625)$

Step 3.2

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

$- 1.3 x \cdot 1 = \ln (0.625)$

Step 3.3

Multiply $- 1.3$ by $1$ .

$- 1.3 x = \ln (0.625)$

Step 4

Divide each term in $- 1.3 x = \ln (0.625)$ by $- 1.3$ and simplify.

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Step 4.1

Divide each term in $- 1.3 x = \ln (0.625)$ by $- 1.3$ .

$\frac{- 1.3 x}{- 1.3} = \frac{\ln (0.625)}{- 1.3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 1.3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 1.3 x}{- 1.3} = \frac{\ln (0.625)}{- 1.3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (0.625)}{- 1.3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.625)}{1.3}$

Step 4.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (0.625)}{1.3}$

Step 4.3.3

Log base $2.71828182$ of $0.625$ is approximately $- 0.47000362$ .

$x = - \frac{- 0.47000362}{1.3}$

Step 4.3.4

Divide $- 0.47000362$ by $1.3$ .

$x = - - 0.36154125$

Step 4.3.5

Multiply $- 1$ by $- 0.36154125$ .

$x = 0.36154125$