Algebra Examples

$\log (1 - x) = \frac{1}{5} \log (\frac{4500}{9000})$

Step 1

Simplify the right side.

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Step 1.1

Combine $\frac{1}{5}$ and $\log (\frac{4500}{9000})$ .

$\log (1 - x) = \frac{\log (\frac{4500}{9000})}{5}$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\log (1 - x) - \frac{\log (\frac{4500}{9000})}{5} = 0$

Step 3

Cancel the common factor of $4500$ and $9000$ .

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Step 3.1

Factor $4500$ out of $4500$ .

$\log (1 - x) - \frac{\log (\frac{4500 (1)}{9000})}{5} = 0$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4500$ out of $9000$ .

$\log (1 - x) - \frac{\log (\frac{4500 \cdot 1}{4500 \cdot 2})}{5} = 0$

Step 3.2.2

Cancel the common factor.

$\log (1 - x) - \frac{\log (\frac{4500 \cdot 1}{4500 \cdot 2})}{5} = 0$

Step 3.2.3

Rewrite the expression.

$\log (1 - x) - \frac{\log (\frac{1}{2})}{5} = 0$

Step 4

To write $\log (1 - x)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\log (1 - x) \cdot \frac{5}{5} - \frac{\log (\frac{1}{2})}{5} = 0$

Step 5

Simplify terms.

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Step 5.1

Combine $\log (1 - x)$ and $\frac{5}{5}$ .

$\frac{\log (1 - x) \cdot 5}{5} - \frac{\log (\frac{1}{2})}{5} = 0$

Step 5.2

Combine the numerators over the common denominator.

$\frac{\log (1 - x) \cdot 5 - \log (\frac{1}{2})}{5} = 0$

Step 6

Move $5$ to the left of $\log (1 - x)$ .

$\frac{5 \log (1 - x) - \log (\frac{1}{2})}{5} = 0$

Step 7

Simplify the left side.

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Step 7.1

Simplify $\frac{5 \log (1 - x) - \log (\frac{1}{2})}{5}$ .

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Step 7.1.1

Simplify the numerator.

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Step 7.1.1.1

Simplify $5 \log (1 - x)$ by moving $5$ inside the logarithm.

$\frac{\log ({(1 - x)}^{5}) - \log (\frac{1}{2})}{5} = 0$

Step 7.1.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\log (\frac{{(1 - x)}^{5}}{\frac{1}{2}})}{5} = 0$

Step 7.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log ({(1 - x)}^{5} \cdot 2)}{5} = 0$

Step 7.1.1.4

Move $2$ to the left of ${(1 - x)}^{5}$ .

$\frac{\log (2 {(1 - x)}^{5})}{5} = 0$

Step 7.1.2

Rewrite $\frac{\log (2 {(1 - x)}^{5})}{5}$ as $\frac{1}{5} \log (2 {(1 - x)}^{5})$ .

$\frac{1}{5} \log (2 {(1 - x)}^{5}) = 0$

Step 7.1.3

Simplify $\frac{1}{5} \log (2 {(1 - x)}^{5})$ by moving $\frac{1}{5}$ inside the logarithm.

$\log ({(2 {(1 - x)}^{5})}^{\frac{1}{5}}) = 0$

Step 7.1.4

Apply the product rule to $2 {(1 - x)}^{5}$ .

$\log (2^{\frac{1}{5}} {({(1 - x)}^{5})}^{\frac{1}{5}}) = 0$

Step 7.1.5

Multiply the exponents in ${({(1 - x)}^{5})}^{\frac{1}{5}}$ .

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Step 7.1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (2^{\frac{1}{5}} {(1 - x)}^{5 (\frac{1}{5})}) = 0$

Step 7.1.5.2

Cancel the common factor of $5$ .

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Step 7.1.5.2.1

Cancel the common factor.

$\log (2^{\frac{1}{5}} {(1 - x)}^{5 (\frac{1}{5})}) = 0$

Step 7.1.5.2.2

Rewrite the expression.

$\log (2^{\frac{1}{5}} {(1 - x)}^{1}) = 0$

Step 7.1.6

Simplify.

$\log (2^{\frac{1}{5}} (1 - x)) = 0$

Step 7.1.7

Apply the distributive property.

$\log (2^{\frac{1}{5}} \cdot 1 + 2^{\frac{1}{5}} (- x)) = 0$

Step 7.1.8

Simplify the expression.

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Step 7.1.8.1

Multiply $2^{\frac{1}{5}}$ by $1$ .

$\log (2^{\frac{1}{5}} + 2^{\frac{1}{5}} (- x)) = 0$

Step 7.1.8.2

Reorder factors in $\log (2^{\frac{1}{5}} + 2^{\frac{1}{5}} (- x))$ .

$\log (2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}}) = 0$

Step 8

Rewrite $\log (2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = 2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}}$

Step 9

Solve for $x$ .

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Step 9.1

Rewrite the equation as $2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}} = 10^{0}$ .

$2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}} = 10^{0}$

Step 9.2

Anything raised to $0$ is $1$ .

$2^{\frac{1}{5}} - x \cdot 2^{\frac{1}{5}} = 1$

Step 9.3

Subtract $2^{\frac{1}{5}}$ from both sides of the equation.

$- x \cdot 2^{\frac{1}{5}} = 1 - 2^{\frac{1}{5}}$

Step 9.4

Divide each term in $- x \cdot 2^{\frac{1}{5}} = 1 - 2^{\frac{1}{5}}$ by $- 2^{\frac{1}{5}}$ and simplify.

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Step 9.4.1

Divide each term in $- x \cdot 2^{\frac{1}{5}} = 1 - 2^{\frac{1}{5}}$ by $- 2^{\frac{1}{5}}$ .

$\frac{- x \cdot 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}} = \frac{1}{- 2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.2

Simplify the left side.

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Step 9.4.2.1

Dividing two negative values results in a positive value.

$\frac{x \cdot 2^{\frac{1}{5}}}{2^{\frac{1}{5}}} = \frac{1}{- 2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.2.2

Cancel the common factor.

$\frac{x \cdot 2^{\frac{1}{5}}}{2^{\frac{1}{5}}} = \frac{1}{- 2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.2.3

Divide $x$ by $1$ .

$x = \frac{1}{- 2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.3

Simplify the right side.

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Step 9.4.3.1

Simplify each term.

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Step 9.4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$x = \frac{- 1 (- 1)}{- 2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.3.1.2

Move the negative in front of the fraction.

$x = - \frac{1}{2^{\frac{1}{5}}} + \frac{- 2^{\frac{1}{5}}}{- 2^{\frac{1}{5}}}$

Step 9.4.3.1.3

Dividing two negative values results in a positive value.

$x = - \frac{1}{2^{\frac{1}{5}}} + \frac{2^{\frac{1}{5}}}{2^{\frac{1}{5}}}$

Step 9.4.3.1.4

Cancel the common factor.

$x = - \frac{1}{2^{\frac{1}{5}}} + \frac{2^{\frac{1}{5}}}{2^{\frac{1}{5}}}$

Step 9.4.3.1.5

Rewrite the expression.

$x = - \frac{1}{2^{\frac{1}{5}}} + 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2^{\frac{1}{5}}} + 1$

Decimal Form:

$x = 0.12944943 \dots$