Finite Math Examples

$500 e^{0.03 x} = 800$

Step 1

Divide each term in $500 e^{0.03 x} = 800$ by $500$ and simplify.

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Step 1.1

Divide each term in $500 e^{0.03 x} = 800$ by $500$ .

$\frac{500 e^{0.03 x}}{500} = \frac{800}{500}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $500$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{500 e^{0.03 x}}{500} = \frac{800}{500}$

Step 1.2.1.2

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

$e^{0.03 x} = \frac{800}{500}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $800$ and $500$ .

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Step 1.3.1.1

Factor $100$ out of $800$ .

$e^{0.03 x} = \frac{100 (8)}{500}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $100$ out of $500$ .

$e^{0.03 x} = \frac{100 \cdot 8}{100 \cdot 5}$

Step 1.3.1.2.2

Cancel the common factor.

$e^{0.03 x} = \frac{100 \cdot 8}{100 \cdot 5}$

Step 1.3.1.2.3

Rewrite the expression.

$e^{0.03 x} = \frac{8}{5}$

Step 2

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

$\ln (e^{0.03 x}) = \ln (\frac{8}{5})$

Step 3

Expand the left side.

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Step 3.1

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

$0.03 x \ln (e) = \ln (\frac{8}{5})$

Step 3.2

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

$0.03 x \cdot 1 = \ln (\frac{8}{5})$

Step 3.3

Multiply $0.03$ by $1$ .

$0.03 x = \ln (\frac{8}{5})$

Step 4

Divide each term in $0.03 x = \ln (\frac{8}{5})$ by $0.03$ and simplify.

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Step 4.1

Divide each term in $0.03 x = \ln (\frac{8}{5})$ by $0.03$ .

$\frac{0.03 x}{0.03} = \frac{\ln (\frac{8}{5})}{0.03}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.03$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.03 x}{0.03} = \frac{\ln (\frac{8}{5})}{0.03}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{8}{5})}{0.03}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (\frac{8}{5})}{0.03}$

Step 4.3.2

Divide $8$ by $5$ .

$x = \frac{\log_{2.71828182} (1.6)}{0.03}$

Step 4.3.3

Log base $2.71828182$ of $1.6$ is approximately $0.47000362$ .

$x = \frac{0.47000362}{0.03}$

Step 4.3.4

Divide $0.47000362$ by $0.03$ .

$x = 15.66678764$