Precalculus Examples

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

Step 1

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

$x^{4} = x^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - x^{2} = 0$

Step 2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - u = 0$

Step 2.2.3

Factor $u$ out of $u^{2} - u$ .

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Step 2.2.3.1

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

$u \cdot u - u = 0$

Step 2.2.3.2

Factor $u$ out of $- u$ .

$u \cdot u + u \cdot - 1 = 0$

Step 2.2.3.3

Factor $u$ out of $u \cdot u + u \cdot - 1$ .

$u (u - 1) = 0$

Step 2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 1) = 0$

Step 2.3

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

$x^{2} = 0$

$x^{2} - 1 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 1$ equal to $0$ .

$x^{2} - 1 = 0$

Step 2.5.2

Solve $x^{2} - 1 = 0$ for $x$ .

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Step 2.5.2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 2.5.2.2

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

$x = \pm \sqrt{1}$

Step 2.5.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 2.6

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

$x = 0, 1, - 1$

Step 3

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

$x = 1, - 1$