Algebra Examples

$\log_{6} (x^{2} + 8) = 1 + \log_{6} (x)$

Step 1

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

$\log_{6} (x^{2} + 8) - \log_{6} (x) = 1$

Step 2

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

$\log_{6} (\frac{x^{2} + 8}{x}) = 1$

Step 3

Rewrite $\log_{6} (\frac{x^{2} + 8}{x}) = 1$ 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 $b^{y} = x$ .

$6^{1} = \frac{x^{2} + 8}{x}$

Step 4

Cross multiply to remove the fraction.

$x^{2} + 8 = 6 (x)$

Step 5

Multiply $6$ by $x$ .

$x^{2} + 8 = 6 x$

Step 6

Subtract $6 x$ from both sides of the equation.

$x^{2} + 8 - 6 x = 0$

Step 7

Factor $x^{2} + 8 - 6 x$ using the AC method.

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Step 7.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 7.2

Write the factored form using these integers.

$(x - 4) (x - 2) = 0$

Step 8

Simplify $(x - 4) (x - 2)$ .

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Step 8.1

Expand $(x - 4) (x - 2)$ using the FOIL Method.

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Step 8.1.1

Apply the distributive property.

$x (x - 2) - 4 (x - 2) = 0$

Step 8.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 4 (x - 2) = 0$

Step 8.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 4 x - 4 \cdot - 2 = 0$

Step 8.2

Simplify and combine like terms.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 4 x - 4 \cdot - 2 = 0$

Step 8.2.1.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 4 x - 4 \cdot - 2 = 0$

Step 8.2.1.3

Multiply $- 4$ by $- 2$ .

$x^{2} - 2 x - 4 x + 8 = 0$

Step 8.2.2

Subtract $4 x$ from $- 2 x$ .

$x^{2} - 6 x + 8 = 0$

Step 9

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 9.2

Write the factored form using these integers.

$(x - 4) (x - 2) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x - 2 = 0$

Step 11

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 11.2

Add $4$ to both sides of the equation.

$x = 4$

Step 12

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 12.2

Add $2$ to both sides of the equation.

$x = 2$

Step 13

The final solution is all the values that make $(x - 4) (x - 2) = 0$ true.

$x = 4, 2$