Algebra Examples

$\log (x - 5) - \log (1 - x) = \frac{1}{3}$

Step 1

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

$\log (\frac{x - 5}{1 - x}) = \frac{1}{3}$

Step 2

Rewrite $\log (\frac{x - 5}{1 - x}) = \frac{1}{3}$ 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 $b^{y} = x$ .

$10^{\frac{1}{3}} = \frac{x - 5}{1 - x}$

Step 3

Cross multiply to remove the fraction.

$x - 5 = 10^{\frac{1}{3}} (1 - x)$

Step 4

Simplify $10^{\frac{1}{3}} (1 - x)$ .

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Step 4.1

Apply the distributive property.

$x - 5 = 10^{\frac{1}{3}} \cdot 1 + 10^{\frac{1}{3}} (- x)$

Step 4.2

Simplify the expression.

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Step 4.2.1

Multiply $10^{\frac{1}{3}}$ by $1$ .

$x - 5 = 10^{\frac{1}{3}} + 10^{\frac{1}{3}} (- x)$

Step 4.2.2

Reorder factors in $10^{\frac{1}{3}} + 10^{\frac{1}{3}} (- x)$ .

$x - 5 = 10^{\frac{1}{3}} - x \cdot 10^{\frac{1}{3}}$

Step 5

Add $x \cdot 10^{\frac{1}{3}}$ to both sides of the equation.

$x - 5 + x \cdot 10^{\frac{1}{3}} = 10^{\frac{1}{3}}$

Step 6

Add $5$ to both sides of the equation.

$x + x \cdot 10^{\frac{1}{3}} = 10^{\frac{1}{3}} + 5$

Step 7

Factor $x$ out of $x + x \cdot 10^{\frac{1}{3}}$ .

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Step 7.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 + x \cdot 10^{\frac{1}{3}} = 10^{\frac{1}{3}} + 5$

Step 7.2

Factor $x$ out of $x \cdot 10^{\frac{1}{3}}$ .

$x \cdot 1 + x (10^{\frac{1}{3}}) = 10^{\frac{1}{3}} + 5$

Step 7.3

Factor $x$ out of $x \cdot 1 + x (10^{\frac{1}{3}})$ .

$x (1 + 10^{\frac{1}{3}}) = 10^{\frac{1}{3}} + 5$

Step 8

Divide each term in $x (1 + 10^{\frac{1}{3}}) = 10^{\frac{1}{3}} + 5$ by $1 + 10^{\frac{1}{3}}$ and simplify.

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Step 8.1

Divide each term in $x (1 + 10^{\frac{1}{3}}) = 10^{\frac{1}{3}} + 5$ by $1 + 10^{\frac{1}{3}}$ .

$\frac{x (1 + 10^{\frac{1}{3}})}{1 + 10^{\frac{1}{3}}} = \frac{10^{\frac{1}{3}}}{1 + 10^{\frac{1}{3}}} + \frac{5}{1 + 10^{\frac{1}{3}}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor.

$\frac{x (1 + 10^{\frac{1}{3}})}{1 + 10^{\frac{1}{3}}} = \frac{10^{\frac{1}{3}}}{1 + 10^{\frac{1}{3}}} + \frac{5}{1 + 10^{\frac{1}{3}}}$

Step 8.2.2

Divide $x$ by $1$ .

$x = \frac{10^{\frac{1}{3}}}{1 + 10^{\frac{1}{3}}} + \frac{5}{1 + 10^{\frac{1}{3}}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$x = \frac{10^{\frac{1}{3}} + 5}{1 + 10^{\frac{1}{3}}}$

Step 9

Exclude the solutions that do not make $\log (x - 5) - \log (1 - x) = \frac{1}{3}$ true.

$No solution$