Algebra Examples

$2 \log (3 x) - \log (9) = 1$

Step 1

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

$2 \log (3 x) - \log (9) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log (3 x) - \log (9)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $2 \log (3 x)$ by moving $2$ inside the logarithm.

$\log ({(3 x)}^{2}) - \log (9) = 1$

Step 2.1.1.2

Apply the product rule to $3 x$ .

$\log (3^{2} x^{2}) - \log (9) = 1$

Step 2.1.1.3

Raise $3$ to the power of $2$ .

$\log (9 x^{2}) - \log (9) = 1$

Step 2.1.2

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

$\log (\frac{9 x^{2}}{9}) = 1$

Step 2.1.3

Cancel the common factor of $9$ .

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Step 2.1.3.1

Cancel the common factor.

$\log (\frac{9 x^{2}}{9}) = 1$

Step 2.1.3.2

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

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

Step 3

Write in exponential form.

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Step 3.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 10$ , $x = x^{2}$ , and $y = 1$ .

$b = 10$

$x = x^{2}$

$y = 1$

Step 3.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$10 = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{2} = 10$ .

$x^{2} = 10$

Step 4.2

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

$x = \pm \sqrt{10}$

Step 4.3

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

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Step 4.3.1

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

$x = \sqrt{10}$

Step 4.3.2

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

$x = - \sqrt{10}$

Step 4.3.3

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

$x = \sqrt{10}, - \sqrt{10}$

Step 5

Exclude the solutions that do not make $2 \log (3 x) - \log (9) = 1$ true.

$x = \sqrt{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{10}$

Decimal Form:

$x = 3.16227766 \dots$