Trigonometry Examples

$\log (2 x^{4}) - \log (x^{2}) = \log (18) + \log (x)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{2 x^{4}}{x^{2}}) = \log (18) + \log (x)$

Step 1.2

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.2.1

Factor $x^{2}$ out of $2 x^{4}$ .

$\log (\frac{x^{2} (2 x^{2})}{x^{2}}) = \log (18) + \log (x)$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Multiply by $1$ .

$\log (\frac{x^{2} (2 x^{2})}{x^{2} \cdot 1}) = \log (18) + \log (x)$

Step 1.2.2.2

Cancel the common factor.

$\log (\frac{x^{2} (2 x^{2})}{x^{2} \cdot 1}) = \log (18) + \log (x)$

Step 1.2.2.3

Rewrite the expression.

$\log (\frac{2 x^{2}}{1}) = \log (18) + \log (x)$

Step 1.2.2.4

Divide $2 x^{2}$ by $1$ .

$\log (2 x^{2}) = \log (18) + \log (x)$

Step 2

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

$\log (2 x^{2}) = \log (18 x)$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$2 x^{2} = 18 x$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $18 x$ from both sides of the equation.

$2 x^{2} - 18 x = 0$

Step 4.2

Factor $2 x$ out of $2 x^{2} - 18 x$ .

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Step 4.2.1

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

$2 x (x) - 18 x = 0$

Step 4.2.2

Factor $2 x$ out of $- 18 x$ .

$2 x (x) + 2 x (- 9) = 0$

Step 4.2.3

Factor $2 x$ out of $2 x (x) + 2 x (- 9)$ .

$2 x (x - 9) = 0$

Step 4.3

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

$x = 0$

$x - 9 = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4.5.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4.6

The final solution is all the values that make $2 x (x - 9) = 0$ true.

$x = 0, 9$

Step 5

Exclude the solutions that do not make $\log (2 x^{4}) - \log (x^{2}) = \log (18) + \log (x)$ true.

$x = 9$