Algebra Examples

$\log_{2} (6 x) - 3 \log_{2} (2) = 0$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{2} (6 x) - 3 \cdot 1 = 0$

Step 1.2

Multiply $- 3$ by $1$ .

$\log_{2} (6 x) - 3 = 0$

Step 2

Add $3$ to both sides of the equation.

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

Step 3

Rewrite $\log_{2} (6 x) = 3$ 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 $b^{y} = x$ .

$2^{3} = 6 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $6 x = 2^{3}$ .

$6 x = 2^{3}$

Step 4.2

Divide each term in $6 x = 2^{3}$ by $6$ and simplify.

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Step 4.2.1

Divide each term in $6 x = 2^{3}$ by $6$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $6$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x = \frac{8}{6}$

Step 4.2.3.2

Cancel the common factor of $8$ and $6$ .

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Step 4.2.3.2.1

Factor $2$ out of $8$ .

$x = \frac{2 (4)}{6}$

Step 4.2.3.2.2

Cancel the common factors.

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Step 4.2.3.2.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 4}{2 \cdot 3}$

Step 4.2.3.2.2.2

Cancel the common factor.

$x = \frac{2 \cdot 4}{2 \cdot 3}$

Step 4.2.3.2.2.3

Rewrite the expression.

$x = \frac{4}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}$

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

$x = 1 \frac{1}{3}$