Precalculus Examples

$4 e^{2 x} = 16 e^{- 6 x}$

Step 1

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

$\ln (4 e^{2 x}) = \ln (16 e^{- 6 x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 e^{2 x})$ as $\ln (4) + \ln (e^{2 x})$ .

$\ln (4) + \ln (e^{2 x}) = \ln (16 e^{- 6 x})$

Step 2.2

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$\ln (4) + 2 x \ln (e) = \ln (16 e^{- 6 x})$

Step 2.3

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

$\ln (4) + 2 x \cdot 1 = \ln (16 e^{- 6 x})$

Step 2.4

Multiply $2$ by $1$ .

$\ln (4) + 2 x = \ln (16 e^{- 6 x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (16 e^{- 6 x})$ as $\ln (16) + \ln (e^{- 6 x})$ .

$\ln (4) + 2 x = \ln (16) + \ln (e^{- 6 x})$

Step 3.2

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

$\ln (4) + 2 x = \ln (16) - 6 x \ln (e)$

Step 3.3

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

$\ln (4) + 2 x = \ln (16) - 6 x \cdot 1$

Step 3.4

Multiply $- 6$ by $1$ .

$\ln (4) + 2 x = \ln (16) - 6 x$

Step 4

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

$\ln (4) - \ln (16) = - 2 x - 6 x$

Step 5

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

$\ln (\frac{4}{16}) = - 2 x - 6 x$

Step 6

Cancel the common factor of $4$ and $16$ .

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Step 6.1

Factor $4$ out of $4$ .

$\ln (\frac{4 (1)}{16}) = - 2 x - 6 x$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $4$ out of $16$ .

$\ln (\frac{4 \cdot 1}{4 \cdot 4}) = - 2 x - 6 x$

Step 6.2.2

Cancel the common factor.

$\ln (\frac{4 \cdot 1}{4 \cdot 4}) = - 2 x - 6 x$

Step 6.2.3

Rewrite the expression.

$\ln (\frac{1}{4}) = - 2 x - 6 x$

Step 7

Subtract $6 x$ from $- 2 x$ .

$\ln (\frac{1}{4}) = - 8 x$

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.

$- 8 x = \ln (\frac{1}{4})$

Step 9

Divide each term in $- 8 x = \ln (\frac{1}{4})$ by $- 8$ and simplify.

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Step 9.1

Divide each term in $- 8 x = \ln (\frac{1}{4})$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{\ln (\frac{1}{4})}{- 8}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 8$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{\ln (\frac{1}{4})}{- 8}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{4})}{- 8}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{4})}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{1}{4})}{8}$

Decimal Form:

$x = 0.17328679 \dots$