Trigonometry Examples

$2 \log_{3} (x) - \log_{3} (4) = \log_{3} (16)$

Step 1

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

$2 \log_{3} (x) - \log_{3} (4) - \log_{3} (16) = 0$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify $2 \log_{3} (x) - \log_{3} (4) - \log_{3} (16)$ .

Tap for more steps...

Step 2.1.1

Simplify $2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{2}) - \log_{3} (4) - \log_{3} (16) = 0$

Step 2.1.2

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

$\log_{3} (\frac{x^{2}}{4}) - \log_{3} (16) = 0$

Step 2.1.3

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

$\log_{3} (\frac{\frac{x^{2}}{4}}{16}) = 0$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (\frac{x^{2}}{4} \cdot \frac{1}{16}) = 0$

Step 2.1.5

Combine.

$\log_{3} (\frac{x^{2} \cdot 1}{4 \cdot 16}) = 0$

Step 2.1.6

Multiply.

Tap for more steps...

Step 2.1.6.1

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

$\log_{3} (\frac{x^{2}}{4 \cdot 16}) = 0$

Step 2.1.6.2

Multiply $4$ by $16$ .

$\log_{3} (\frac{x^{2}}{64}) = 0$

Step 3

Rewrite $\log_{3} (\frac{x^{2}}{64}) = 0$ 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 $b^{y} = x$ .

$3^{0} = \frac{x^{2}}{64}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{x^{2}}{64} = 3^{0}$ .

$\frac{x^{2}}{64} = 3^{0}$

Step 4.2

Multiply both sides of the equation by $64$ .

$64 \frac{x^{2}}{64} = 64 \cdot 3^{0}$

Step 4.3

Simplify both sides of the equation.

Tap for more steps...

Step 4.3.1

Simplify the left side.

Tap for more steps...

Step 4.3.1.1

Cancel the common factor of $64$ .

Tap for more steps...

Step 4.3.1.1.1

Cancel the common factor.

$64 \frac{x^{2}}{64} = 64 \cdot 3^{0}$

Step 4.3.1.1.2

Rewrite the expression.

$x^{2} = 64 \cdot 3^{0}$

Step 4.3.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.1

Simplify $64 \cdot 3^{0}$ .

Tap for more steps...

Step 4.3.2.1.1

Anything raised to $0$ is $1$ .

$x^{2} = 64 \cdot 1$

Step 4.3.2.1.2

Multiply $64$ by $1$ .

$x^{2} = 64$

Step 4.4

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

$x = \pm \sqrt{64}$

Step 4.5

Simplify $\pm \sqrt{64}$ .

Tap for more steps...

Step 4.5.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 4.5.2

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

$x = \pm 8$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$x = 8$

Step 4.6.2

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

$x = - 8$

Step 4.6.3

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

$x = 8, - 8$

Step 5

Exclude the solutions that do not make $2 \log_{3} (x) - \log_{3} (4) = \log_{3} (16)$ true.

$x = 8$