Precalculus Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln (2^{- 4 x + 1})$ by moving $- 4 x + 1$ outside the logarithm.

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(- 4 x + 1) \ln (2)$ .

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Subtract $2 x \ln (3)$ from both sides of the equation.

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

Factor $2 x$ out of $2 x (- 2 \ln (2)) + 2 x (- 1 \ln (3))$ .

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

Step 4.6

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $2$ .

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Rewrite the expression.

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

Step 4.7.2.2

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

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Step 4.7.2.2.1

Cancel the common factor.

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

Step 4.7.2.2.2

Divide $x$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Simplify each term.

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Step 4.7.3.1.1

Move the negative in front of the fraction.

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

Step 4.7.3.1.2

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.80264263 \dots$