Precalculus Examples

$3^{x + 4} = 2^{1 - 3 x}$

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (3^{x + 4})$ by moving $x + 4$ outside the logarithm.

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

Step 3

Expand $\ln (2^{1 - 3 x})$ by moving $1 - 3 x$ outside the logarithm.

$(x + 4) \ln (3) = (1 - 3 x) \ln (2)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 4) \ln (3)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x + 4) \ln (3) = (1 - 3 x) \ln (2)$

Step 4.1.2

Simplify by adding zeros.

$(x + 4) \ln (3) = (1 - 3 x) \ln (2)$

Step 4.1.3

Apply the distributive property.

$x \ln (3) + 4 \ln (3) = (1 - 3 x) \ln (2)$

Step 4.2

Simplify $(1 - 3 x) \ln (2)$ .

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Step 4.2.1

Apply the distributive property.

$x \ln (3) + 4 \ln (3) = 1 \ln (2) - 3 x \ln (2)$

Step 4.2.2

Multiply $\ln (2)$ by $1$ .

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

Step 4.3

Add $3 x \ln (2)$ to both sides of the equation.

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

Step 4.4

Subtract $4 \ln (3)$ from both sides of the equation.

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

Step 4.5

Factor $x$ out of $x \ln (3) + 3 x \ln (2)$ .

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Step 4.5.1

Factor $x$ out of $x \ln (3)$ .

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

Step 4.5.2

Factor $x$ out of $3 x \ln (2)$ .

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

Step 4.5.3

Factor $x$ out of $x (\ln (3)) + x (3 \ln (2))$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $\ln (3) + 3 \ln (2)$ .

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Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.16464420 \dots$