Solve for x 2 log of x- log of 3 = log of 3

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

Step 1

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

$2 \log (x) - \log (3) - \log (3) = 0$

Step 2

Subtract $\log (3)$ from $- \log (3)$ .

$2 \log (x) - 2 \log (3) = 0$

Step 3

Simplify the left side.

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

Simplify each term.

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

$\log (x^{2}) - 2 \log (3) = 0$

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

$\log (x^{2}) - \log (3^{2}) = 0$

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

$\log (x^{2}) - \log (9) = 0$

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

$\log (\frac{x^{2}}{9}) = 0$

Step 4

Rewrite $\log (\frac{x^{2}}{9}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = \frac{x^{2}}{9}$

Step 5

Solve for $x$ .

Rewrite the equation as $\frac{x^{2}}{9} = 10^{0}$ .

$\frac{x^{2}}{9} = 10^{0}$

Multiply both sides of the equation by $9$ .

$9 \frac{x^{2}}{9} = 9 \cdot 10^{0}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$9 \frac{x^{2}}{9} = 9 \cdot 10^{0}$

Rewrite the expression.

$x^{2} = 9 \cdot 10^{0}$

Simplify the right side.

Convert $9 \cdot 10^{0}$ from scientific notation.

$x^{2} = 9$

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

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

$x = \pm 3$

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

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

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

Step 6

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

$x = 3$

Solve for x 2 log of x- log of 3 = log of 3

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