Finite Math Examples

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

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

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})

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

Step 2

Expand the left side.

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Step 2.1

Rewrite

ln (12 e^{6.8 x})

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

ln (12) + ln (e^{6.8 x})

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

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

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

Step 2.2

Expand

ln (e^{6.8 x})

$\ln (e^{6.8 x})$ by moving

6.8 x

$6.8 x$ outside the logarithm.

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

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

Step 2.3

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 2.4

Multiply

6.8

$6.8$ by

1

$1$ .

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

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

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

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

Step 3

Expand the right side.

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Step 3.1

Rewrite

ln (8 e^{3 x})

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

ln (8) + ln (e^{3 x})

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

ln (12) + 6.8 x = 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})

$\ln (e^{3 x})$ by moving

3 x

$3 x$ outside the logarithm.

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

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

Step 3.3

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 3.4

Multiply

3

$3$ by

1

$1$ .

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

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

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

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

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

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

Step 5

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

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

Step 6

Cancel the common factor of

12

$12$ and

8

$8$ .

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Step 6.1

Factor

4

$4$ out of

12

$12$ .

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor

4

$4$ out of

8

$8$ .

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

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

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

$0.10670134$ .

$x = - 0.10670134$