Algebra Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

Expand $\ln ({(\frac{5}{9})}^{5 x - 9})$ by moving $5 x - 9$ outside the logarithm.

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

Step 3

Rewrite $\ln (\frac{5}{9})$ as $\ln (5) - \ln (9)$ .

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

Step 4

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

Expand $\ln ({(\frac{9}{5})}^{9 x - 5})$ by moving $9 x - 5$ outside the logarithm.

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

Step 8

Rewrite $\ln (\frac{9}{5})$ as $\ln (9) - \ln (5)$ .

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

Step 9

Solve the equation for $x$ .

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Step 9.1

Simplify $(5 x - 9) (\ln (5) - 2 \ln (3))$ .

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Step 9.1.1

Rewrite.

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

Step 9.1.2

Simplify by adding zeros.

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

Step 9.1.3

Expand $(5 x - 9) (\ln (5) - 2 \ln (3))$ using the FOIL Method.

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Step 9.1.3.1

Apply the distributive property.

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

Step 9.1.3.2

Apply the distributive property.

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

Step 9.1.3.3

Apply the distributive property.

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

Step 9.1.4

Simplify each term.

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Step 9.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 9.1.4.2

Multiply $5$ by $- 2$ .

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) - 9 (- 2 \ln (3)) = (9 x - 5) (\ln (9) - \ln (5))$

Step 9.1.4.3

Multiply $- 2$ by $- 9$ .

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = (9 x - 5) (\ln (9) - \ln (5))$

Step 9.2

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

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Step 9.2.1

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

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Step 9.2.1.1

Apply the distributive property.

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x (\ln (9) - \ln (5)) - 5 (\ln (9) - \ln (5))$

Step 9.2.1.2

Apply the distributive property.

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x \ln (9) + 9 x (- \ln (5)) - 5 (\ln (9) - \ln (5))$

Step 9.2.1.3

Apply the distributive property.

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x \ln (9) + 9 x (- \ln (5)) - 5 \ln (9) - 5 (- \ln (5))$

Step 9.2.2

Simplify each term.

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Step 9.2.2.1

Rewrite using the commutative property of multiplication.

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x \ln (9) + 9 \cdot - 1 x \ln (5) - 5 \ln (9) - 5 (- \ln (5))$

Step 9.2.2.2

Multiply $9$ by $- 1$ .

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x \ln (9) - 9 x \ln (5) - 5 \ln (9) - 5 (- \ln (5))$

Step 9.2.2.3

Multiply $- 1$ by $- 5$ .

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) = 9 x \ln (9) - 9 x \ln (5) - 5 \ln (9) + 5 \ln (5)$

Step 9.3

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

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Step 9.3.1

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

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) - 9 x \ln (9) = - 9 x \ln (5) - 5 \ln (9) + 5 \ln (5)$

Step 9.3.2

Add $9 x \ln (5)$ to both sides of the equation.

$5 x \ln (5) - 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) - 9 x \ln (9) + 9 x \ln (5) = - 5 \ln (9) + 5 \ln (5)$

Step 9.3.3

Add $5 x \ln (5)$ and $9 x \ln (5)$ .

$- 10 x \ln (3) - 9 \ln (5) + 18 \ln (3) - 9 x \ln (9) + 14 x \ln (5) = - 5 \ln (9) + 5 \ln (5)$

Step 9.4

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

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Step 9.4.1

Add $9 \ln (5)$ to both sides of the equation.

$- 10 x \ln (3) + 18 \ln (3) - 9 x \ln (9) + 14 x \ln (5) = - 5 \ln (9) + 5 \ln (5) + 9 \ln (5)$

Step 9.4.2

Subtract $18 \ln (3)$ from both sides of the equation.

$- 10 x \ln (3) - 9 x \ln (9) + 14 x \ln (5) = - 5 \ln (9) + 5 \ln (5) + 9 \ln (5) - 18 \ln (3)$

Step 9.4.3

Add $5 \ln (5)$ and $9 \ln (5)$ .

$- 10 x \ln (3) - 9 x \ln (9) + 14 x \ln (5) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.5

Factor $x$ out of $- 10 x \ln (3) - 9 x \ln (9) + 14 x \ln (5)$ .

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Step 9.5.1

Factor $x$ out of $- 10 x \ln (3)$ .

$x (- 10 \ln (3)) - 9 x \ln (9) + 14 x \ln (5) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.5.2

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

$x (- 10 \ln (3)) + x (- 9 \ln (9)) + 14 x \ln (5) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.5.3

Factor $x$ out of $14 x \ln (5)$ .

$x (- 10 \ln (3)) + x (- 9 \ln (9)) + x (14 \ln (5)) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.5.4

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

$x (- 10 \ln (3) - 9 \ln (9)) + x (14 \ln (5)) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.5.5

Factor $x$ out of $x (- 10 \ln (3) - 9 \ln (9)) + x (14 \ln (5))$ .

$x (- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$

Step 9.6

Divide each term in $x (- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$ by $- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)$ and simplify.

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Step 9.6.1

Divide each term in $x (- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)) = - 5 \ln (9) + 14 \ln (5) - 18 \ln (3)$ by $- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)$ .

$\frac{x (- 10 \ln (3) - 9 \ln (9) + 14 \ln (5))}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} = \frac{- 5 \ln (9)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{14 \ln (5)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{- 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.2

Simplify the left side.

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Step 9.6.2.1

Cancel the common factor of $- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)$ .

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Step 9.6.2.1.1

Cancel the common factor.

Step 9.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (9)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{14 \ln (5)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{- 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3

Simplify the right side.

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Step 9.6.3.1

Simplify each term.

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Step 9.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{5 \ln (9)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{14 \ln (5)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{- 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5 \ln (9)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} + \frac{14 \ln (5)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} - \frac{18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2

Simplify terms.

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Step 9.6.3.2.1

Combine the numerators over the common denominator.

$x = \frac{- 5 \ln (9) + 14 \ln (5)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)} - \frac{18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.2

Combine the numerators over the common denominator.

$x = \frac{- 5 \ln (9) + 14 \ln (5) - 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.3

Factor $- 1$ out of $- 5 \ln (9)$ .

$x = \frac{- (5 \ln (9)) + 14 \ln (5) - 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.4

Factor $- 1$ out of $14 \ln (5)$ .

$x = \frac{- (5 \ln (9)) - (- 14 \ln (5)) - 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.5

Factor $- 1$ out of $- (5 \ln (9)) - (- 14 \ln (5))$ .

$x = \frac{- (5 \ln (9) - 14 \ln (5)) - 18 \ln (3)}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.6

Factor $- 1$ out of $- 18 \ln (3)$ .

$x = \frac{- (5 \ln (9) - 14 \ln (5)) - (18 \ln (3))}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.7

Factor $- 1$ out of $- (5 \ln (9) - 14 \ln (5)) - (18 \ln (3))$ .

$x = \frac{- (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.8

Rewrite $- (5 \ln (9) - 14 \ln (5) + 18 \ln (3))$ as $- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))$ .

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- 10 \ln (3) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.9

Factor $- 1$ out of $- 10 \ln (3)$ .

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- (10 \ln (3)) - 9 \ln (9) + 14 \ln (5)}$

Step 9.6.3.2.10

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

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- (10 \ln (3)) - (9 \ln (9)) + 14 \ln (5)}$

Step 9.6.3.2.11

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

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- (10 \ln (3) + 9 \ln (9)) + 14 \ln (5)}$

Step 9.6.3.2.12

Factor $- 1$ out of $14 \ln (5)$ .

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- (10 \ln (3) + 9 \ln (9)) - (- 14 \ln (5))}$

Step 9.6.3.2.13

Factor $- 1$ out of $- (10 \ln (3) + 9 \ln (9)) - (- 14 \ln (5))$ .

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- (10 \ln (3) + 9 \ln (9) - 14 \ln (5))}$

Step 9.6.3.2.14

Rewrite $- (10 \ln (3) + 9 \ln (9) - 14 \ln (5))$ as $- 1 (10 \ln (3) + 9 \ln (9) - 14 \ln (5))$ .

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- 1 (10 \ln (3) + 9 \ln (9) - 14 \ln (5))}$

Step 9.6.3.2.15

Cancel the common factor.

$x = \frac{- 1 (5 \ln (9) - 14 \ln (5) + 18 \ln (3))}{- 1 (10 \ln (3) + 9 \ln (9) - 14 \ln (5))}$

Step 9.6.3.2.16

Rewrite the expression.

$x = \frac{5 \ln (9) - 14 \ln (5) + 18 \ln (3)}{10 \ln (3) + 9 \ln (9) - 14 \ln (5)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \ln (9) - 14 \ln (5) + 18 \ln (3)}{10 \ln (3) + 9 \ln (9) - 14 \ln (5)}$

Decimal Form:

$x = 1$