Algebra Examples

3 \log_{9} (x^{2}) = 6 \log_{9} (x)

$3 \log_{9} (x^{2}) = 6 \log_{9} (x)$

Step 1

Simplify.

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Step 1.1

Simplify

3 \log_{9} (x^{2})

$3 \log_{9} (x^{2})$ by moving

3

$3$ inside the logarithm.

\log_{9} ({(x^{2})}^{3}) = 6 \log_{9} (x)

$\log_{9} ({(x^{2})}^{3}) = 6 \log_{9} (x)$

Step 1.2

Simplify

6 \log_{9} (x)

$6 \log_{9} (x)$ by moving

6

$6$ inside the logarithm.

\log_{9} ({(x^{2})}^{3}) = \log_{9} (x^{6})

$\log_{9} ({(x^{2})}^{3}) = \log_{9} (x^{6})$

\log_{9} ({(x^{2})}^{3}) = \log_{9} (x^{6})

$\log_{9} ({(x^{2})}^{3}) = \log_{9} (x^{6})$

Step 2

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

{(x^{2})}^{3} = x^{6}

${(x^{2})}^{3} = x^{6}$

Step 3

Solve for

x

$x$ .

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Step 3.1

Multiply the exponents in

{(x^{2})}^{3}

${(x^{2})}^{3}$ .

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Step 3.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

x^{2 \cdot 3} = x^{6}

$x^{2 \cdot 3} = x^{6}$

Step 3.1.2

Multiply

2

$2$ by

3

$3$ .

x^{6} = x^{6}

$x^{6} = x^{6}$

x^{6} = x^{6}

$x^{6} = x^{6}$

Step 3.2

Move all terms containing

x

$x$ to the left side of the equation.

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Step 3.2.1

Subtract

x^{6}

$x^{6}$ from both sides of the equation.

x^{6} - x^{6} = 0

$x^{6} - x^{6} = 0$

Step 3.2.2

Subtract

x^{6}

$x^{6}$ from

x^{6}

$x^{6}$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 3.3

Since

0 = 0

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

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$