Algebra Examples

$\log_{3} (x) + \log_{3} (x^{2} + 2) = 1 + 2 \log_{3} (x)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{3} (x (x^{2} + 2)) = 1 + 2 \log_{3} (x)$

Step 1.2

Apply the distributive property.

$\log_{3} (x \cdot x^{2} + x \cdot 2) = 1 + 2 \log_{3} (x)$

Step 1.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.3.1

Multiply $x$ by $x^{2}$ .

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Step 1.3.1.1

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

$\log_{3} (x \cdot x^{2} + x \cdot 2) = 1 + 2 \log_{3} (x)$

Step 1.3.1.2

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

$\log_{3} (x^{1 + 2} + x \cdot 2) = 1 + 2 \log_{3} (x)$

Step 1.3.2

Add $1$ and $2$ .

$\log_{3} (x^{3} + x \cdot 2) = 1 + 2 \log_{3} (x)$

Step 1.4

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

$\log_{3} (x^{3} + 2 x) = 1 + 2 \log_{3} (x)$

Step 2

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

$\log_{3} (x^{3} + 2 x) - 2 \log_{3} (x) = 1$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\log_{3} (x^{3} + 2 x) - 2 \log_{3} (x)$ .

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Step 3.1.1

Simplify $- 2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{3} + 2 x) - \log_{3} (x^{2}) = 1$

Step 3.1.2

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

$\log_{3} (\frac{x^{3} + 2 x}{x^{2}}) = 1$

Step 3.1.3

Factor $x$ out of $x^{3} + 2 x$ .

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Step 3.1.3.1

Factor $x$ out of $x^{3}$ .

$\log_{3} (\frac{x \cdot x^{2} + 2 x}{x^{2}}) = 1$

Step 3.1.3.2

Factor $x$ out of $2 x$ .

$\log_{3} (\frac{x \cdot x^{2} + x \cdot 2}{x^{2}}) = 1$

Step 3.1.3.3

Factor $x$ out of $x \cdot x^{2} + x \cdot 2$ .

$\log_{3} (\frac{x (x^{2} + 2)}{x^{2}}) = 1$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $x$ out of $x^{2}$ .

$\log_{3} (\frac{x (x^{2} + 2)}{x \cdot x}) = 1$

Step 3.1.4.2

Cancel the common factor.

$\log_{3} (\frac{x (x^{2} + 2)}{x \cdot x}) = 1$

Step 3.1.4.3

Rewrite the expression.

$\log_{3} (\frac{x^{2} + 2}{x}) = 1$

Step 4

Rewrite $\log_{3} (\frac{x^{2} + 2}{x}) = 1$ 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 $b^{y} = x$ .

$3^{1} = \frac{x^{2} + 2}{x}$

Step 5

Cross multiply to remove the fraction.

$x^{2} + 2 = 3 (x)$

Step 6

Multiply $3$ by $x$ .

$x^{2} + 2 = 3 x$

Step 7

Subtract $3 x$ from both sides of the equation.

$x^{2} + 2 - 3 x = 0$

Step 8

Factor $x^{2} + 2 - 3 x$ using the AC method.

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Step 8.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 8.2

Write the factored form using these integers.

$(x - 2) (x - 1) = 0$

Step 9

Simplify $(x - 2) (x - 1)$ .

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Step 9.1

Expand $(x - 2) (x - 1)$ using the FOIL Method.

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Step 9.1.1

Apply the distributive property.

$x (x - 1) - 2 (x - 1) = 0$

Step 9.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 2 (x - 1) = 0$

Step 9.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 2 x - 2 \cdot - 1 = 0$

Step 9.2

Simplify and combine like terms.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 2 x - 2 \cdot - 1 = 0$

Step 9.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 9.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 9.2.1.4

Multiply $- 2$ by $- 1$ .

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

Step 9.2.2

Subtract $2 x$ from $- x$ .

$x^{2} - 3 x + 2 = 0$

Step 10

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 10.1

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 10.2

Write the factored form using these integers.

$(x - 2) (x - 1) = 0$

Step 11

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

$x - 2 = 0$

$x - 1 = 0$

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

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 13.2

Add $1$ to both sides of the equation.

$x = 1$

Step 14

The final solution is all the values that make $(x - 2) (x - 1) = 0$ true.

$x = 2, 1$