Algebra Examples

${(\frac{2}{3})}^{5 x + 2} = {(\frac{3}{2})}^{2 x}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{2}{3})}^{5 x + 2}) = \ln ({(\frac{3}{2})}^{2 x})$

Step 2

Expand $\ln ({(\frac{2}{3})}^{5 x + 2})$ by moving $5 x + 2$ outside the logarithm.

$(5 x + 2) \ln (\frac{2}{3}) = \ln ({(\frac{3}{2})}^{2 x})$

Step 3

Rewrite $\ln (\frac{2}{3})$ as $\ln (2) - \ln (3)$ .

$(5 x + 2) (\ln (2) - \ln (3)) = \ln ({(\frac{3}{2})}^{2 x})$

Step 4

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

$(5 x + 2) (\ln (2) - \ln (3)) = 2 x \ln (\frac{3}{2})$

Step 5

Rewrite $\ln (\frac{3}{2})$ as $\ln (3) - \ln (2)$ .

$(5 x + 2) (\ln (2) - \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6

Solve the equation for $x$ .

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Step 6.1

Simplify $(5 x + 2) (\ln (2) - \ln (3))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + (5 x + 2) (\ln (2) - \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.2

Simplify by adding zeros.

$(5 x + 2) (\ln (2) - \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.3

Expand $(5 x + 2) (\ln (2) - \ln (3))$ using the FOIL Method.

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Step 6.1.3.1

Apply the distributive property.

$5 x (\ln (2) - \ln (3)) + 2 (\ln (2) - \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.3.2

Apply the distributive property.

$5 x \ln (2) + 5 x (- \ln (3)) + 2 (\ln (2) - \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.3.3

Apply the distributive property.

$5 x \ln (2) + 5 x (- \ln (3)) + 2 \ln (2) + 2 (- \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.4

Simplify each term.

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Step 6.1.4.1

Rewrite using the commutative property of multiplication.

$5 x \ln (2) + 5 \cdot - 1 x \ln (3) + 2 \ln (2) + 2 (- \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.4.2

Multiply $5$ by $- 1$ .

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) + 2 (- \ln (3)) = 2 x (\ln (3) - \ln (2))$

Step 6.1.4.3

Multiply $- 1$ by $2$ .

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) = 2 x (\ln (3) - \ln (2))$

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) = 2 x \ln (3) + 2 x (- \ln (2))$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Rewrite using the commutative property of multiplication.

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) = 2 x \ln (3) + 2 \cdot - 1 x \ln (2)$

Step 6.2.2.2

Multiply $2$ by $- 1$ .

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) = 2 x \ln (3) - 2 x \ln (2)$

Step 6.3

Move all terms containing $x$ to the left side of the equation.

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Step 6.3.1

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

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) - 2 x \ln (3) = - 2 x \ln (2)$

Step 6.3.2

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

$5 x \ln (2) - 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) - 2 x \ln (3) + 2 x \ln (2) = 0$

Step 6.3.3

Add $5 x \ln (2)$ and $2 x \ln (2)$ .

$- 5 x \ln (3) + 2 \ln (2) - 2 \ln (3) - 2 x \ln (3) + 7 x \ln (2) = 0$

Step 6.3.4

Subtract $2 x \ln (3)$ from $- 5 x \ln (3)$ .

$2 \ln (2) - 2 \ln (3) - 7 x \ln (3) + 7 x \ln (2) = 0$

Step 6.4

Move all terms not containing $x$ to the right side of the equation.

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Step 6.4.1

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

$- 2 \ln (3) - 7 x \ln (3) + 7 x \ln (2) = - 2 \ln (2)$

Step 6.4.2

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

$- 7 x \ln (3) + 7 x \ln (2) = - 2 \ln (2) + 2 \ln (3)$

Step 6.5

Factor $7 x$ out of $- 7 x \ln (3) + 7 x \ln (2)$ .

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Step 6.5.1

Factor $7 x$ out of $- 7 x \ln (3)$ .

$7 x (- 1 \ln (3)) + 7 x \ln (2) = - 2 \ln (2) + 2 \ln (3)$

Step 6.5.2

Factor $7 x$ out of $7 x \ln (2)$ .

$7 x (- 1 \ln (3)) + 7 x \ln (2) = - 2 \ln (2) + 2 \ln (3)$

Step 6.5.3

Factor $7 x$ out of $7 x (- 1 \ln (3)) + 7 x \ln (2)$ .

$7 x (- 1 \ln (3) + \ln (2)) = - 2 \ln (2) + 2 \ln (3)$

Step 6.6

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

$7 x (- \ln (3) + \ln (2)) = - 2 \ln (2) + 2 \ln (3)$

Step 6.7

Divide each term in $7 x (- \ln (3) + \ln (2)) = - 2 \ln (2) + 2 \ln (3)$ by $7 (- \ln (3) + \ln (2))$ and simplify.

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Step 6.7.1

Divide each term in $7 x (- \ln (3) + \ln (2)) = - 2 \ln (2) + 2 \ln (3)$ by $7 (- \ln (3) + \ln (2))$ .

$\frac{7 x (- \ln (3) + \ln (2))}{7 (- \ln (3) + \ln (2))} = \frac{- 2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 6.7.2

Simplify the left side.

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Step 6.7.2.1

Cancel the common factor of $7$ .

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Step 6.7.2.1.1

Cancel the common factor.

$\frac{7 x (- \ln (3) + \ln (2))}{7 (- \ln (3) + \ln (2))} = \frac{- 2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 6.7.2.1.2

Rewrite the expression.

$\frac{x (- \ln (3) + \ln (2))}{- \ln (3) + \ln (2)} = \frac{- 2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 6.7.2.2

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

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Step 6.7.2.2.1

Cancel the common factor.

$\frac{x (- \ln (3) + \ln (2))}{- \ln (3) + \ln (2)} = \frac{- 2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 6.7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 6.7.3

Simplify the right side.

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Step 6.7.3.1

Move the negative in front of the fraction.

$x = - \frac{2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2 \ln (2)}{7 (- \ln (3) + \ln (2))} + \frac{2 \ln (3)}{7 (- \ln (3) + \ln (2))}$

Decimal Form:

$x = - 0.\overline{285714}$