Finite Math Examples

$12 e^{6.8 x} = 8 e^{3 x}$

Step 1

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

$\ln (12 e^{6.8 x}) = \ln (8 e^{3 x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (12 e^{6.8 x})$ as $\ln (12) + \ln (e^{6.8 x})$ .

$\ln (12) + \ln (e^{6.8 x}) = \ln (8 e^{3 x})$

Step 2.2

Expand $\ln (e^{6.8 x})$ by moving $6.8 x$ outside the logarithm.

$\ln (12) + 6.8 x \ln (e) = \ln (8 e^{3 x})$

Step 2.3

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

$\ln (12) + 6.8 x \cdot 1 = \ln (8 e^{3 x})$

Step 2.4

Multiply $6.8$ by $1$ .

$\ln (12) + 6.8 x = \ln (8 e^{3 x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (8 e^{3 x})$ as $\ln (8) + \ln (e^{3 x})$ .

$\ln (12) + 6.8 x = \ln (8) + \ln (e^{3 x})$

Step 3.2

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

$\ln (12) + 6.8 x = \ln (8) + 3 x \ln (e)$

Step 3.3

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

$\ln (12) + 6.8 x = \ln (8) + 3 x \cdot 1$

Step 3.4

Multiply $3$ by $1$ .

$\ln (12) + 6.8 x = \ln (8) + 3 x$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$\ln (12) - \ln (8) = - 6.8 x + 3 x$

Step 5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{12}{8}) = - 6.8 x + 3 x$

Step 6

Cancel the common factor of $12$ and $8$ .

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Step 6.1

Factor $4$ out of $12$ .

$\ln (\frac{4 (3)}{8}) = - 6.8 x + 3 x$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $4$ out of $8$ .

$\ln (\frac{4 \cdot 3}{4 \cdot 2}) = - 6.8 x + 3 x$

Step 6.2.2

Cancel the common factor.

$\ln (\frac{4 \cdot 3}{4 \cdot 2}) = - 6.8 x + 3 x$

Step 6.2.3

Rewrite the expression.

$\ln (\frac{3}{2}) = - 6.8 x + 3 x$

Step 7

Add $- 6.8 x$ and $3 x$ .

$\ln (\frac{3}{2}) = - 3.8 x$

Step 8

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 3.8 x = \ln (\frac{3}{2})$

Step 9

Divide each term in $- 3.8 x = \ln (\frac{3}{2})$ by $- 3.8$ and simplify.

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Step 9.1

Divide each term in $- 3.8 x = \ln (\frac{3}{2})$ by $- 3.8$ .

$\frac{- 3.8 x}{- 3.8} = \frac{\ln (\frac{3}{2})}{- 3.8}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 3.8$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 3.8 x}{- 3.8} = \frac{\ln (\frac{3}{2})}{- 3.8}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{3}{2})}{- 3.8}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{3}{2})}{3.8}$

Step 9.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (\frac{3}{2})}{3.8}$

Step 9.3.3

Divide $3$ by $2$ .

$x = - \frac{\log_{2.71828182} (1.5)}{3.8}$

Step 9.3.4

Log base $2.71828182$ of $1.5$ is approximately $0.4054651$ .

$x = - \frac{0.4054651}{3.8}$

Step 9.3.5

Divide $0.4054651$ by $3.8$ .

$x = - 1 \cdot 0.10670134$

Step 9.3.6

Multiply $- 1$ by $0.10670134$ .

$x = - 0.10670134$