Precalculus Examples

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

Step 1

Simplify

$2 \log (x)$ by moving

$2$ inside the logarithm.

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

Step 2

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

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

Step 3

Solve for

$x$ .

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Step 3.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | x |$

Step 3.2

Solve for

$x$ .

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Step 3.2.1

Rewrite the absolute value equation as four equations without absolute value bars.

$x = x$

$x = - x$

$- x = x$

$- x = - x$

Step 3.2.2

After simplifying, there are only two unique equations to be solved.

$x = x$

$x = - x$

Step 3.2.3

Solve

$x = x$ for

$x$ .

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Step 3.2.3.1

Move all terms containing

$x$ to the left side of the equation.

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Step 3.2.3.1.1

Subtract

$x$ from both sides of the equation.

$x - x = 0$

Step 3.2.3.1.2

Subtract

$x$ from

$x$ .

$0 = 0$

Step 3.2.3.2

Since

$0 = 0$ , the equation will always be true.

All real numbers

Step 3.2.4

Solve

$x = - x$ for

$x$ .

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Step 3.2.4.1

Move all terms containing

$x$ to the left side of the equation.

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Step 3.2.4.1.1

Add

$x$ to both sides of the equation.

$x + x = 0$

Step 3.2.4.1.2

Add

$x$ and

$x$ .

$2 x = 0$

Step 3.2.4.2

Divide each term in

$2 x = 0$ by

$2$ and simplify.

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Step 3.2.4.2.1

Divide each term in

$2 x = 0$ by

$2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of

$2$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.4.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{0}{2}$

Step 3.2.4.2.3

Simplify the right side.

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Step 3.2.4.2.3.1

Divide

$0$ by

$2$ .

$x = 0$

Step 3.2.5

List all of the solutions.

$x = 0$

Step 4

Exclude the solutions that do not make

$\log ({(x)}^{2}) = 2 \log (x)$ true.

$No solution$