Precalculus Examples

$4^{x + 2} = 6^{- 2 x - 3}$

Step 1

Take the log of both sides of the equation.

$\ln (4^{x + 2}) = \ln (6^{- 2 x - 3})$

Step 2

Expand $\ln (4^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (4) = \ln (6^{- 2 x - 3})$

Step 3

Expand $\ln (6^{- 2 x - 3})$ by moving $- 2 x - 3$ outside the logarithm.

$(x + 2) \ln (4) = (- 2 x - 3) \ln (6)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 2) \ln (4)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x + 2) \ln (4) = (- 2 x - 3) \ln (6)$

Step 4.1.2

Simplify by adding zeros.

$(x + 2) \ln (4) = (- 2 x - 3) \ln (6)$

Step 4.1.3

Apply the distributive property.

$x \ln (4) + 2 \ln (4) = (- 2 x - 3) \ln (6)$

Step 4.2

Apply the distributive property.

$x \ln (4) + 2 \ln (4) = - 2 x \ln (6) - 3 \ln (6)$

Step 4.3

Add $2 x \ln (6)$ to both sides of the equation.

$x \ln (4) + 2 \ln (4) + 2 x \ln (6) = - 3 \ln (6)$

Step 4.4

Subtract $2 \ln (4)$ from both sides of the equation.

$x \ln (4) + 2 x \ln (6) = - 3 \ln (6) - 2 \ln (4)$

Step 4.5

Factor $x$ out of $x \ln (4) + 2 x \ln (6)$ .

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Step 4.5.1

Factor $x$ out of $x \ln (4)$ .

$x (\ln (4)) + 2 x \ln (6) = - 3 \ln (6) - 2 \ln (4)$

Step 4.5.2

Factor $x$ out of $2 x \ln (6)$ .

$x (\ln (4)) + x (2 \ln (6)) = - 3 \ln (6) - 2 \ln (4)$

Step 4.5.3

Factor $x$ out of $x (\ln (4)) + x (2 \ln (6))$ .

$x (\ln (4) + 2 \ln (6)) = - 3 \ln (6) - 2 \ln (4)$

Step 4.6

Divide each term in $x (\ln (4) + 2 \ln (6)) = - 3 \ln (6) - 2 \ln (4)$ by $\ln (4) + 2 \ln (6)$ and simplify.

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Step 4.6.1

Divide each term in $x (\ln (4) + 2 \ln (6)) = - 3 \ln (6) - 2 \ln (4)$ by $\ln (4) + 2 \ln (6)$ .

$\frac{x (\ln (4) + 2 \ln (6))}{\ln (4) + 2 \ln (6)} = \frac{- 3 \ln (6)}{\ln (4) + 2 \ln (6)} + \frac{- 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (4) + 2 \ln (6)$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (4) + 2 \ln (6))}{\ln (4) + 2 \ln (6)} = \frac{- 3 \ln (6)}{\ln (4) + 2 \ln (6)} + \frac{- 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (6)}{\ln (4) + 2 \ln (6)} + \frac{- 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

$x = \frac{- 3 \ln (6) - 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 4.6.3.2

Factor $- 1$ out of $- 3 \ln (6)$ .

$x = \frac{- (3 \ln (6)) - 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 4.6.3.3

Factor $- 1$ out of $- 2 \ln (4)$ .

$x = \frac{- (3 \ln (6)) - (2 \ln (4))}{\ln (4) + 2 \ln (6)}$

Step 4.6.3.4

Factor $- 1$ out of $- (3 \ln (6)) - (2 \ln (4))$ .

$x = \frac{- (3 \ln (6) + 2 \ln (4))}{\ln (4) + 2 \ln (6)}$

Step 4.6.3.5

Simplify the expression.

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Step 4.6.3.5.1

Rewrite $- (3 \ln (6) + 2 \ln (4))$ as $- 1 (3 \ln (6) + 2 \ln (4))$ .

$x = \frac{- 1 (3 \ln (6) + 2 \ln (4))}{\ln (4) + 2 \ln (6)}$

Step 4.6.3.5.2

Move the negative in front of the fraction.

$x = - \frac{3 \ln (6) + 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \ln (6) + 2 \ln (4)}{\ln (4) + 2 \ln (6)}$

Decimal Form:

$x = - 1.63947147 \dots$