Solve for x 2 log base 2 of 2+2 log base 2 of 6- log base 2 of 3x=3

$2 \log_{2} (2) + 2 \log_{2} (6) - \log_{2} (3 x) = 3$

Simplify each term.

Logarithm base $2$ of $2$ is $1$ .

$2 \cdot 1 + 2 \log_{2} (6) - \log_{2} (3 x) = 3$

Multiply $2$ by $1$ .

$2 + 2 \log_{2} (6) - \log_{2} (3 x) = 3$

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

$2 \log_{2} (6) - \log_{2} (3 x) = - 2 + 3$

Add $- 2$ and $3$ .

$2 \log_{2} (6) - \log_{2} (3 x) = 1$

Simplify the left side.

Simplify $2 \log_{2} (6) - \log_{2} (3 x)$ .

Simplify each term.

Simplify $2 \log_{2} (6)$ by moving $2$ inside the logarithm.

$\log_{2} (6^{2}) - \log_{2} (3 x) = 1$

Raise $6$ to the power of $2$ .

$\log_{2} (36) - \log_{2} (3 x) = 1$

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

$\log_{2} (\frac{36}{3 x}) = 1$

Cancel the common factor of $36$ and $3$ .

Factor $3$ out of $36$ .

$\log_{2} (\frac{3 \cdot 12}{3 x}) = 1$

Cancel the common factors.

Factor $3$ out of $3 x$ .

$\log_{2} (\frac{3 \cdot 12}{3 (x)}) = 1$

Cancel the common factor.

$\log_{2} (\frac{3 \cdot 12}{3 x}) = 1$

Rewrite the expression.

$\log_{2} (\frac{12}{x}) = 1$

Rewrite $\log_{2} (\frac{12}{x}) = 1$ 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 $b^{y} = x$ .

$2^{1} = \frac{12}{x}$

Solve for $x$ .

Rewrite the equation as $\frac{12}{x} = 2^{1}$ .

$\frac{12}{x} = 2$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $\frac{12}{x} = 2$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{12}{x} = 2$ by $x$ .

$\frac{12}{x} x = 2 x$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{12}{x} x = 2 x$

Rewrite the expression.

$12 = 2 x$

Solve the equation.

Rewrite the equation as $2 x = 12$ .

$2 x = 12$

Divide each term in $2 x = 12$ by $2$ and simplify.

Divide each term in $2 x = 12$ by $2$ .

$\frac{2 x}{2} = \frac{12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{12}{2}$

Divide $x$ by $1$ .

$x = \frac{12}{2}$

Simplify the right side.

Divide $12$ by $2$ .

$x = 6$

Solve for x 2 log base 2 of 2+2 log base 2 of 6- log base 2 of 3x=3

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