Algebra Examples

${(\frac{5}{6})}^{4 x} = {(\frac{36}{25})}^{9 - x}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{5}{6})}^{4 x}) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 2

Expand $\ln ({(\frac{5}{6})}^{4 x})$ by moving $4 x$ outside the logarithm.

$4 x \ln (\frac{5}{6}) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 3

Rewrite $\ln (\frac{5}{6})$ as $\ln (5) - \ln (6)$ .

$4 x (\ln (5) - \ln (6)) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 4

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$4 x (\ln (5) - \ln (2 (3))) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 5

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$4 x (\ln (5) - (\ln (2) + \ln (3))) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 6

Apply the distributive property.

$4 x (\ln (5) - \ln (2) - \ln (3)) = \ln ({(\frac{36}{25})}^{9 - x})$

Step 7

Expand $\ln ({(\frac{36}{25})}^{9 - x})$ by moving $9 - x$ outside the logarithm.

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) \ln (\frac{36}{25})$

Step 8

Rewrite $\ln (\frac{36}{25})$ as $\ln (36) - \ln (25)$ .

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - \ln (25))$

Step 9

Rewrite $\ln (25)$ as $\ln (5^{2})$ .

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - \ln (5^{2}))$

Step 10

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

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - (2 \ln (5)))$

Step 11

Multiply $2$ by $- 1$ .

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12

Solve the equation for $x$ .

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Step 12.1

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

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Step 12.1.1

Rewrite.

$0 + 0 + 4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.2

Simplify by adding zeros.

$4 x (\ln (5) - \ln (2) - \ln (3)) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.3

Apply the distributive property.

$4 x \ln (5) + 4 x (- \ln (2)) + 4 x (- \ln (3)) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.4

Simplify.

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Step 12.1.4.1

Rewrite using the commutative property of multiplication.

$4 x \ln (5) + 4 \cdot - 1 x \ln (2) + 4 x (- \ln (3)) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.4.2

Rewrite using the commutative property of multiplication.

$4 x \ln (5) + 4 \cdot - 1 x \ln (2) + 4 \cdot - 1 x \ln (3) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.5

Simplify each term.

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Step 12.1.5.1

Multiply $4$ by $- 1$ .

$4 x \ln (5) - 4 x \ln (2) + 4 \cdot - 1 x \ln (3) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.1.5.2

Multiply $4$ by $- 1$ .

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = (9 - x) (\ln (36) - 2 \ln (5))$

Step 12.2

Simplify $(9 - x) (\ln (36) - 2 \ln (5))$ .

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Step 12.2.1

Expand $(9 - x) (\ln (36) - 2 \ln (5))$ using the FOIL Method.

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Step 12.2.1.1

Apply the distributive property.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 (\ln (36) - 2 \ln (5)) - x (\ln (36) - 2 \ln (5))$

Step 12.2.1.2

Apply the distributive property.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 \ln (36) + 9 (- 2 \ln (5)) - x (\ln (36) - 2 \ln (5))$

Step 12.2.1.3

Apply the distributive property.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 \ln (36) + 9 (- 2 \ln (5)) - x \ln (36) - x (- 2 \ln (5))$

Step 12.2.2

Simplify each term.

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Step 12.2.2.1

Multiply $- 2$ by $9$ .

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 \ln (36) - 18 \ln (5) - x \ln (36) - x (- 2 \ln (5))$

Step 12.2.2.2

Rewrite using the commutative property of multiplication.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 \ln (36) - 18 \ln (5) - x \ln (36) - 1 \cdot - 2 x \ln (5)$

Step 12.2.2.3

Multiply $- 1$ by $- 2$ .

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) = 9 \ln (36) - 18 \ln (5) - x \ln (36) + 2 x \ln (5)$

Step 12.3

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

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Step 12.3.1

Add $x \ln (36)$ to both sides of the equation.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) + x \ln (36) = 9 \ln (36) - 18 \ln (5) + 2 x \ln (5)$

Step 12.3.2

Subtract $2 x \ln (5)$ from both sides of the equation.

$4 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) + x \ln (36) - 2 x \ln (5) = 9 \ln (36) - 18 \ln (5)$

Step 12.3.3

Subtract $2 x \ln (5)$ from $4 x \ln (5)$ .

$2 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) + x \ln (36) = 9 \ln (36) - 18 \ln (5)$

Step 12.4

Factor $x$ out of $2 x \ln (5) - 4 x \ln (2) - 4 x \ln (3) + x \ln (36)$ .

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Step 12.4.1

Factor $x$ out of $2 x \ln (5)$ .

$x (2 \ln (5)) - 4 x \ln (2) - 4 x \ln (3) + x \ln (36) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.2

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

$x (2 \ln (5)) + x (- 4 \ln (2)) - 4 x \ln (3) + x \ln (36) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.3

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

$x (2 \ln (5)) + x (- 4 \ln (2)) + x (- 4 \ln (3)) + x \ln (36) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.4

Factor $x$ out of $x \ln (36)$ .

$x (2 \ln (5)) + x (- 4 \ln (2)) + x (- 4 \ln (3)) + x (\ln (36)) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.5

Factor $x$ out of $x (2 \ln (5)) + x (- 4 \ln (2))$ .

$x (2 \ln (5) - 4 \ln (2)) + x (- 4 \ln (3)) + x (\ln (36)) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.6

Factor $x$ out of $x (2 \ln (5) - 4 \ln (2)) + x (- 4 \ln (3))$ .

$x (2 \ln (5) - 4 \ln (2) - 4 \ln (3)) + x (\ln (36)) = 9 \ln (36) - 18 \ln (5)$

Step 12.4.7

Factor $x$ out of $x (2 \ln (5) - 4 \ln (2) - 4 \ln (3)) + x (\ln (36))$ .

$x (2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)) = 9 \ln (36) - 18 \ln (5)$

Step 12.5

Divide each term in $x (2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)) = 9 \ln (36) - 18 \ln (5)$ by $2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)$ and simplify.

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Step 12.5.1

Divide each term in $x (2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)) = 9 \ln (36) - 18 \ln (5)$ by $2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)$ .

$\frac{x (2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36))}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)} = \frac{9 \ln (36)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)} + \frac{- 18 \ln (5)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 12.5.2

Simplify the left side.

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Step 12.5.2.1

Cancel the common factor of $2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)$ .

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Step 12.5.2.1.1

Cancel the common factor.

Step 12.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{9 \ln (36)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)} + \frac{- 18 \ln (5)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 12.5.3

Simplify the right side.

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Step 12.5.3.1

Combine the numerators over the common denominator.

$x = \frac{9 \ln (36) - 18 \ln (5)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 12.5.3.2

Factor $9$ out of $9 \ln (36) - 18 \ln (5)$ .

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Step 12.5.3.2.1

Factor $9$ out of $9 \ln (36)$ .

$x = \frac{9 \ln (36) - 18 \ln (5)}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 12.5.3.2.2

Factor $9$ out of $- 18 \ln (5)$ .

$x = \frac{9 \ln (36) + 9 (- 2 \ln (5))}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 12.5.3.2.3

Factor $9$ out of $9 \ln (36) + 9 (- 2 \ln (5))$ .

$x = \frac{9 (\ln (36) - 2 \ln (5))}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 (\ln (36) - 2 \ln (5))}{2 \ln (5) - 4 \ln (2) - 4 \ln (3) + \ln (36)}$

Decimal Form:

$x = - 9.000000000000 \dots$