Precalculus Examples

$7 e^{6 x} = 4 e^{9 - 3 x}$

Step 1

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

$\ln (7 e^{6 x}) = \ln (4 e^{9 - 3 x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (7 e^{6 x})$ as $\ln (7) + \ln (e^{6 x})$ .

$\ln (7) + \ln (e^{6 x}) = \ln (4 e^{9 - 3 x})$

Step 2.2

Expand $\ln (e^{6 x})$ by moving $6 x$ outside the logarithm.

$\ln (7) + 6 x \ln (e) = \ln (4 e^{9 - 3 x})$

Step 2.3

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

$\ln (7) + 6 x \cdot 1 = \ln (4 e^{9 - 3 x})$

Step 2.4

Multiply $6$ by $1$ .

$\ln (7) + 6 x = \ln (4 e^{9 - 3 x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (4 e^{9 - 3 x})$ as $\ln (4) + \ln (e^{9 - 3 x})$ .

$\ln (7) + 6 x = \ln (4) + \ln (e^{9 - 3 x})$

Step 3.2

Expand $\ln (e^{9 - 3 x})$ by moving $9 - 3 x$ outside the logarithm.

$\ln (7) + 6 x = \ln (4) + (9 - 3 x) \ln (e)$

Step 3.3

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

$\ln (7) + 6 x = \ln (4) + (9 - 3 x) \cdot 1$

Step 3.4

Multiply $9 - 3 x$ by $1$ .

$\ln (7) + 6 x = \ln (4) + 9 - 3 x$

Step 4

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

$\ln (7) - \ln (4) = - 6 x + 9 - 3 x$

Step 5

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

$\ln (\frac{7}{4}) = - 6 x + 9 - 3 x$

Step 6

Subtract $3 x$ from $- 6 x$ .

$\ln (\frac{7}{4}) = - 9 x + 9$

Step 7

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

$- 9 x + 9 = \ln (\frac{7}{4})$

Step 8

Subtract $9$ from both sides of the equation.

$- 9 x = \ln (\frac{7}{4}) - 9$

Step 9

Divide each term in $- 9 x = \ln (\frac{7}{4}) - 9$ by $- 9$ and simplify.

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Step 9.1

Divide each term in $- 9 x = \ln (\frac{7}{4}) - 9$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{\ln (\frac{7}{4})}{- 9} + \frac{- 9}{- 9}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 9$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{\ln (\frac{7}{4})}{- 9} + \frac{- 9}{- 9}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{7}{4})}{- 9} + \frac{- 9}{- 9}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{7}{4})}{9} + \frac{- 9}{- 9}$

Step 9.3.1.2

Divide $- 9$ by $- 9$ .

$x = - \frac{\ln (\frac{7}{4})}{9} + 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{7}{4})}{9} + 1$

Decimal Form:

$x = 0.93782046 \dots$