Algebra Examples

$\log_{3} (18 x^{3}) - \log_{3} (2 x) = \log_{3} (144)$

Step 1

Subtract $\log_{3} (144)$ from both sides of the equation.

$\log_{3} (18 x^{3}) - \log_{3} (2 x) - \log_{3} (144) = 0$

Step 2

Simplify $\log_{3} (18 x^{3}) - \log_{3} (2 x) - \log_{3} (144)$ .

Tap for more steps...

Step 2.1

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

$\log_{3} (\frac{18 x^{3}}{2 x}) - \log_{3} (144) = 0$

Step 2.2

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

$\log_{3} (\frac{\frac{18 x^{3}}{2 x}}{144}) = 0$

Step 2.3

Cancel the common factor of $18$ and $2$ .

Tap for more steps...

Step 2.3.1

Factor $2$ out of $18 x^{3}$ .

$\log_{3} (\frac{\frac{2 (9 x^{3})}{2 x}}{144}) = 0$

Step 2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.1

Factor $2$ out of $2 x$ .

$\log_{3} (\frac{\frac{2 (9 x^{3})}{2 (x)}}{144}) = 0$

Step 2.3.2.2

Cancel the common factor.

$\log_{3} (\frac{\frac{2 (9 x^{3})}{2 x}}{144}) = 0$

Step 2.3.2.3

Rewrite the expression.

$\log_{3} (\frac{\frac{9 x^{3}}{x}}{144}) = 0$

Step 2.4

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 2.4.1

Factor $x$ out of $9 x^{3}$ .

$\log_{3} (\frac{\frac{x (9 x^{2})}{x}}{144}) = 0$

Step 2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.1

Raise $x$ to the power of $1$ .

$\log_{3} (\frac{\frac{x (9 x^{2})}{x^{1}}}{144}) = 0$

Step 2.4.2.2

Factor $x$ out of $x^{1}$ .

$\log_{3} (\frac{\frac{x (9 x^{2})}{x \cdot 1}}{144}) = 0$

Step 2.4.2.3

Cancel the common factor.

$\log_{3} (\frac{\frac{x (9 x^{2})}{x \cdot 1}}{144}) = 0$

Step 2.4.2.4

Rewrite the expression.

$\log_{3} (\frac{\frac{9 x^{2}}{1}}{144}) = 0$

Step 2.4.2.5

Divide $9 x^{2}$ by $1$ .

$\log_{3} (\frac{9 x^{2}}{144}) = 0$

Step 2.5

Cancel the common factor of $9$ and $144$ .

Tap for more steps...

Step 2.5.1

Factor $9$ out of $9 x^{2}$ .

$\log_{3} (\frac{9 (x^{2})}{144}) = 0$

Step 2.5.2

Cancel the common factors.

Tap for more steps...

Step 2.5.2.1

Factor $9$ out of $144$ .

$\log_{3} (\frac{9 x^{2}}{9 \cdot 16}) = 0$

Step 2.5.2.2

Cancel the common factor.

$\log_{3} (\frac{9 x^{2}}{9 \cdot 16}) = 0$

Step 2.5.2.3

Rewrite the expression.

$\log_{3} (\frac{x^{2}}{16}) = 0$

Step 3

Set the argument in $\log_{3} (\frac{x^{2}}{16})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{x^{2}}{16} \leq 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Multiply both sides by $16$ .

$\frac{x^{2}}{16} \cdot 16 \leq 0 \cdot 16$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

Cancel the common factor of $16$ .

Tap for more steps...

Step 4.2.1.1.1

Cancel the common factor.

$\frac{x^{2}}{16} \cdot 16 \leq 0 \cdot 16$

Step 4.2.1.1.2

Rewrite the expression.

$x^{2} \leq 0 \cdot 16$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Multiply $0$ by $16$ .

$x^{2} \leq 0$

Step 4.3

Solve for $x$ .

Tap for more steps...

Step 4.3.1

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

$\sqrt{x^{2}} \leq \sqrt{0}$

Step 4.3.2

Simplify the equation.

Tap for more steps...

Step 4.3.2.1

Simplify the left side.

Tap for more steps...

Step 4.3.2.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{0}$

Step 4.3.2.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.2.1

Simplify $\sqrt{0}$ .

Tap for more steps...

Step 4.3.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| x | \leq \sqrt{0^{2}}$

Step 4.3.2.2.1.2

Pull terms out from under the radical.

$| x | \leq | 0 |$

Step 4.3.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x | \leq 0$

Step 4.3.3

Write $| x | \leq 0$ as a piecewise.

Tap for more steps...

Step 4.3.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 4.3.3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 0$

Step 4.3.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 4.3.3.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 0$

Step 4.3.3.5

Write as a piecewise.

${\begin{cases} x \leq 0 & x \geq 0 \\ - x \leq 0 & x < 0 \end{cases}$

Step 4.3.4

Find the intersection of $x \leq 0$ and $x \geq 0$ .

$x = 0$

Step 4.3.5

Solve $- x \leq 0$ when $x < 0$ .

Tap for more steps...

Step 4.3.5.1

Divide each term in $- x \leq 0$ by $- 1$ and simplify.

Tap for more steps...

Step 4.3.5.1.1

Divide each term in $- x \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{0}{- 1}$

Step 4.3.5.1.2

Simplify the left side.

Tap for more steps...

Step 4.3.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{0}{- 1}$

Step 4.3.5.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{0}{- 1}$

Step 4.3.5.1.3

Simplify the right side.

Tap for more steps...

Step 4.3.5.1.3.1

Divide $0$ by $- 1$ .

$x \geq 0$

Step 4.3.5.2

Find the intersection of $x \geq 0$ and $x < 0$ .

No solution

Step 4.3.6

Find the union of the solutions.

$x = 0$

Step 5