Algebra Examples

${(\frac{2}{3})}^{x - 5} = {(\frac{9}{4})}^{\frac{3 x}{4}}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{2}{3})}^{x - 5}) = \ln ({(\frac{9}{4})}^{\frac{3 x}{4}})$

Step 2

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

$(x - 5) \ln (\frac{2}{3}) = \ln ({(\frac{9}{4})}^{\frac{3 x}{4}})$

Step 3

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

$(x - 5) (\ln (2) - \ln (3)) = \ln ({(\frac{9}{4})}^{\frac{3 x}{4}})$

Step 4

Expand $\ln ({(\frac{9}{4})}^{\frac{3 x}{4}})$ by moving $\frac{3 x}{4}$ outside the logarithm.

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x}{4} \ln (\frac{9}{4})$

Step 5

Combine $\frac{3 x}{4}$ and $\ln (\frac{9}{4})$ .

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x \ln (\frac{9}{4})}{4}$

Step 6

Rewrite $\ln (\frac{9}{4})$ as $\ln (9) - \ln (4)$ .

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x (\ln (9) - \ln (4))}{4}$

Step 7

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

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x (\ln (9) - \ln (2^{2}))}{4}$

Step 8

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

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x (\ln (9) - (2 \ln (2)))}{4}$

Step 9

Multiply $2$ by $- 1$ .

$(x - 5) (\ln (2) - \ln (3)) = \frac{3 x (\ln (9) - 2 \ln (2))}{4}$

Step 10

Solve the equation for $x$ .

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Step 10.1

Multiply both sides by $4$ .

$(x - 5) (\ln (2) - \ln (3)) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2

Simplify.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Simplify $(x - 5) (\ln (2) - \ln (3)) \cdot 4$ .

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Step 10.2.1.1.1

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

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Step 10.2.1.1.1.1

Apply the distributive property.

$(x (\ln (2) - \ln (3)) - 5 (\ln (2) - \ln (3))) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.1.2

Apply the distributive property.

$(x \ln (2) + x (- \ln (3)) - 5 (\ln (2) - \ln (3))) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.1.3

Apply the distributive property.

$(x \ln (2) + x (- \ln (3)) - 5 \ln (2) - 5 (- \ln (3))) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.2

Simplify terms.

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Step 10.2.1.1.2.1

Simplify each term.

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Step 10.2.1.1.2.1.1

Rewrite using the commutative property of multiplication.

$(x \ln (2) - x \ln (3) - 5 \ln (2) - 5 (- \ln (3))) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.2.1.2

Multiply $- 1$ by $- 5$ .

$(x \ln (2) - x \ln (3) - 5 \ln (2) + 5 \ln (3)) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.2.2

Apply the distributive property.

$x \ln (2) \cdot 4 - x \ln (3) \cdot 4 - 5 \ln (2) \cdot 4 + 5 \ln (3) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.3

Simplify.

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Step 10.2.1.1.3.1

Move $4$ to the left of $x \ln (2)$ .

$4 \cdot (x \ln (2)) - x \ln (3) \cdot 4 - 5 \ln (2) \cdot 4 + 5 \ln (3) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.3.2

Multiply $4$ by $- 1$ .

$4 \cdot (x \ln (2)) - 4 x \ln (3) - 5 \ln (2) \cdot 4 + 5 \ln (3) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.3.3

Multiply $4$ by $- 5$ .

$4 \cdot (x \ln (2)) - 4 x \ln (3) - 20 \ln (2) + 5 \ln (3) \cdot 4 = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.3.4

Multiply $4$ by $5$ .

$4 \cdot (x \ln (2)) - 4 x \ln (3) - 20 \ln (2) + 20 \ln (3) = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.4

Remove parentheses.

$4 x \ln (2) - 4 x \ln (3) - 20 \ln (2) + 20 \ln (3) = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.5

Simplify the expression.

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Step 10.2.1.1.5.1

Move $- 20 \ln (2)$ .

$4 x \ln (2) - 4 x \ln (3) + 20 \ln (3) - 20 \ln (2) = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.1.1.5.2

Reorder $4 x \ln (2)$ and $- 4 x \ln (3)$ .

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$ .

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Step 10.2.2.1.1

Cancel the common factor of $4$ .

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Step 10.2.2.1.1.1

Cancel the common factor.

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = \frac{3 x (\ln (9) - 2 \ln (2))}{4} \cdot 4$

Step 10.2.2.1.1.2

Rewrite the expression.

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = 3 x (\ln (9) - 2 \ln (2))$

Step 10.2.2.1.2

Apply the distributive property.

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = 3 x \ln (9) + 3 x (- 2 \ln (2))$

Step 10.2.2.1.3

Simplify the expression.

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Step 10.2.2.1.3.1

Rewrite using the commutative property of multiplication.

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = 3 x \ln (9) + 3 \cdot - 2 x \ln (2)$

Step 10.2.2.1.3.2

Multiply $3$ by $- 2$ .

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) = 3 x \ln (9) - 6 x \ln (2)$

Step 10.3

Solve for $x$ .

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Step 10.3.1

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

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Step 10.3.1.1

Subtract $3 x \ln (9)$ from both sides of the equation.

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) - 3 x \ln (9) = - 6 x \ln (2)$

Step 10.3.1.2

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

$- 4 x \ln (3) + 4 x \ln (2) + 20 \ln (3) - 20 \ln (2) - 3 x \ln (9) + 6 x \ln (2) = 0$

Step 10.3.1.3

Add $4 x \ln (2)$ and $6 x \ln (2)$ .

$- 4 x \ln (3) + 20 \ln (3) - 20 \ln (2) - 3 x \ln (9) + 10 x \ln (2) = 0$

Step 10.3.2

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

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Step 10.3.2.1

Subtract $20 \ln (3)$ from both sides of the equation.

$- 4 x \ln (3) - 20 \ln (2) - 3 x \ln (9) + 10 x \ln (2) = - 20 \ln (3)$

Step 10.3.2.2

Add $20 \ln (2)$ to both sides of the equation.

$- 4 x \ln (3) - 3 x \ln (9) + 10 x \ln (2) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.3

Factor $x$ out of $- 4 x \ln (3) - 3 x \ln (9) + 10 x \ln (2)$ .

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Step 10.3.3.1

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

$x (- 4 \ln (3)) - 3 x \ln (9) + 10 x \ln (2) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.3.2

Factor $x$ out of $- 3 x \ln (9)$ .

$x (- 4 \ln (3)) + x (- 3 \ln (9)) + 10 x \ln (2) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.3.3

Factor $x$ out of $10 x \ln (2)$ .

$x (- 4 \ln (3)) + x (- 3 \ln (9)) + x (10 \ln (2)) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.3.4

Factor $x$ out of $x (- 4 \ln (3)) + x (- 3 \ln (9))$ .

$x (- 4 \ln (3) - 3 \ln (9)) + x (10 \ln (2)) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.3.5

Factor $x$ out of $x (- 4 \ln (3) - 3 \ln (9)) + x (10 \ln (2))$ .

$x (- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)) = - 20 \ln (3) + 20 \ln (2)$

Step 10.3.4

Divide each term in $x (- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)) = - 20 \ln (3) + 20 \ln (2)$ by $- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)$ and simplify.

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Step 10.3.4.1

Divide each term in $x (- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)) = - 20 \ln (3) + 20 \ln (2)$ by $- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)$ .

$\frac{x (- 4 \ln (3) - 3 \ln (9) + 10 \ln (2))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)} = \frac{- 20 \ln (3)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)} + \frac{20 \ln (2)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.2

Simplify the left side.

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Step 10.3.4.2.1

Cancel the common factor of $- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)$ .

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Step 10.3.4.2.1.1

Cancel the common factor.

Step 10.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20 \ln (3)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)} + \frac{20 \ln (2)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3

Simplify the right side.

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Step 10.3.4.3.1

Combine the numerators over the common denominator.

$x = \frac{- 20 \ln (3) + 20 \ln (2)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.2

Factor $20$ out of $- 20 \ln (3) + 20 \ln (2)$ .

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Step 10.3.4.3.2.1

Factor $20$ out of $- 20 \ln (3)$ .

$x = \frac{20 (- \ln (3)) + 20 \ln (2)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.2.2

Factor $20$ out of $20 \ln (2)$ .

$x = \frac{20 (- \ln (3)) + 20 \ln (2)}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.2.3

Factor $20$ out of $20 (- \ln (3)) + 20 \ln (2)$ .

$x = \frac{20 (- \ln (3) + \ln (2))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.3

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

$x = \frac{20 (- \ln (3) + \ln (2))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.4

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

$x = \frac{20 (- \ln (3) - 1 (- \ln (2)))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.5

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

$x = \frac{20 (- (\ln (3) - \ln (2)))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.6

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

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- 4 \ln (3) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.7

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

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- (4 \ln (3)) - 3 \ln (9) + 10 \ln (2)}$

Step 10.3.4.3.8

Factor $- 1$ out of $- 3 \ln (9)$ .

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- (4 \ln (3)) - (3 \ln (9)) + 10 \ln (2)}$

Step 10.3.4.3.9

Factor $- 1$ out of $- (4 \ln (3)) - (3 \ln (9))$ .

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- (4 \ln (3) + 3 \ln (9)) + 10 \ln (2)}$

Step 10.3.4.3.10

Factor $- 1$ out of $10 \ln (2)$ .

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- (4 \ln (3) + 3 \ln (9)) - (- 10 \ln (2))}$

Step 10.3.4.3.11

Factor $- 1$ out of $- (4 \ln (3) + 3 \ln (9)) - (- 10 \ln (2))$ .

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- (4 \ln (3) + 3 \ln (9) - 10 \ln (2))}$

Step 10.3.4.3.12

Rewrite $- (4 \ln (3) + 3 \ln (9) - 10 \ln (2))$ as $- 1 (4 \ln (3) + 3 \ln (9) - 10 \ln (2))$ .

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- 1 (4 \ln (3) + 3 \ln (9) - 10 \ln (2))}$

Step 10.3.4.3.13

Cancel the common factor.

$x = \frac{20 (- 1 (\ln (3) - \ln (2)))}{- 1 (4 \ln (3) + 3 \ln (9) - 10 \ln (2))}$

Step 10.3.4.3.14

Rewrite the expression.

$x = \frac{20 (\ln (3) - \ln (2))}{4 \ln (3) + 3 \ln (9) - 10 \ln (2)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{20 (\ln (3) - \ln (2))}{4 \ln (3) + 3 \ln (9) - 10 \ln (2)}$

Decimal Form:

$x = 2$