Algebra Examples

$\log_{x} (x) + 6 = 2$

Step 1

Move all terms not containing $\log_{x} (x)$ to the right side of the equation.

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Step 1.1

Subtract $6$ from both sides of the equation.

$\log_{x} (x) = 2 - 6$

Step 1.2

Subtract $6$ from $2$ .

$\log_{x} (x) = - 4$

Step 2

Rewrite $\log_{x} (x) = - 4$ 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 $b^{y} = x$ .

$x^{- 4} = x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{4}} = x$

Step 3.2

Subtract $x$ from both sides of the equation.

$\frac{1}{x^{4}} - x = 0$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{4}, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$x^{4}$

Step 3.4

Multiply each term in $\frac{1}{x^{4}} - x = 0$ by $x^{4}$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{1}{x^{4}} - x = 0$ by $x^{4}$ .

$\frac{1}{x^{4}} x^{4} - x \cdot x^{4} = 0 x^{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Cancel the common factor of $x^{4}$ .

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Step 3.4.2.1.1.1

Cancel the common factor.

$\frac{1}{x^{4}} x^{4} - x \cdot x^{4} = 0 x^{4}$

Step 3.4.2.1.1.2

Rewrite the expression.

$1 - x \cdot x^{4} = 0 x^{4}$

Step 3.4.2.1.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 3.4.2.1.2.1

Move $x^{4}$ .

$1 - (x^{4} x) = 0 x^{4}$

Step 3.4.2.1.2.2

Multiply $x^{4}$ by $x$ .

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Step 3.4.2.1.2.2.1

Raise $x$ to the power of $1$ .

$1 - (x^{4} x^{1}) = 0 x^{4}$

Step 3.4.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - x^{4 + 1} = 0 x^{4}$

Step 3.4.2.1.2.3

Add $4$ and $1$ .

$1 - x^{5} = 0 x^{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply $0$ by $x^{4}$ .

$1 - x^{5} = 0$

Step 3.5

Solve the equation.

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Step 3.5.1

Subtract $1$ from both sides of the equation.

$- x^{5} = - 1$

Step 3.5.2

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

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Step 3.5.2.1

Divide each term in $- x^{5} = - 1$ by $- 1$ .

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2.2

Divide $x^{5}$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 1$ by $- 1$ .

$x^{5} = 1$

Step 3.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{1}$

Step 3.5.4

Any root of $1$ is $1$ .

$x = 1$

Step 4

Exclude the solutions that do not make $\log_{x} (x) + 6 = 2$ true.

$No solution$