Algebra Examples

$2 \log (4) - \log (3) + 2 \log (x) - 4 = 0$

Step 1

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

$2 \log (4) - \log (3) + 2 \log (x) = 4 + 0$

Step 2

Add $4$ and $0$ .

$2 \log (4) - \log (3) + 2 \log (x) = 4$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $2 \log (4) - \log (3) + 2 \log (x)$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Simplify $2 \log (4)$ by moving $2$ inside the logarithm.

$\log (4^{2}) - \log (3) + 2 \log (x) = 4$

Step 3.1.1.2

Raise $4$ to the power of $2$ .

$\log (16) - \log (3) + 2 \log (x) = 4$

Step 3.1.1.3

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

$\log (16) - \log (3) + \log (x^{2}) = 4$

Step 3.1.2

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

$\log (\frac{16}{3}) + \log (x^{2}) = 4$

Step 3.1.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 3.1.4

Combine $\frac{16}{3}$ and $x^{2}$ .

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

Step 4

Rewrite $\log (\frac{16 x^{2}}{3}) = 4$ 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 $b^{y} = x$ .

$10^{4} = \frac{16 x^{2}}{3}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $\frac{16 x^{2}}{3} = 10^{4}$ .

$\frac{16 x^{2}}{3} = 10^{4}$

Step 5.2

Multiply both sides of the equation by $\frac{3}{16}$ .

$\frac{3}{16} \cdot \frac{16 x^{2}}{3} = \frac{3}{16} \cdot 10^{4}$

Step 5.3

Simplify both sides of the equation.

Tap for more steps...

Step 5.3.1

Simplify the left side.

Tap for more steps...

Step 5.3.1.1

Simplify $\frac{3}{16} \cdot \frac{16 x^{2}}{3}$ .

Tap for more steps...

Step 5.3.1.1.1

Combine.

$\frac{3 (16 x^{2})}{16 \cdot 3} = \frac{3}{16} \cdot 10^{4}$

Step 5.3.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.1.1.2.1

Cancel the common factor.

$\frac{3 (16 x^{2})}{16 \cdot 3} = \frac{3}{16} \cdot 10^{4}$

Step 5.3.1.1.2.2

Rewrite the expression.

$\frac{16 x^{2}}{16} = \frac{3}{16} \cdot 10^{4}$

Step 5.3.1.1.3

Cancel the common factor of $16$ .

Tap for more steps...

Step 5.3.1.1.3.1

Cancel the common factor.

$\frac{16 x^{2}}{16} = \frac{3}{16} \cdot 10^{4}$

Step 5.3.1.1.3.2

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

$x^{2} = \frac{3}{16} \cdot 10^{4}$

Step 5.3.2

Simplify the right side.

Tap for more steps...

Step 5.3.2.1

Simplify $\frac{3}{16} \cdot 10^{4}$ .

Tap for more steps...

Step 5.3.2.1.1

Raise $10$ to the power of $4$ .

$x^{2} = \frac{3}{16} \cdot 10000$

Step 5.3.2.1.2

Cancel the common factor of $16$ .

Tap for more steps...

Step 5.3.2.1.2.1

Factor $16$ out of $10000$ .

$x^{2} = \frac{3}{16} \cdot (16 (625))$

Step 5.3.2.1.2.2

Cancel the common factor.

$x^{2} = \frac{3}{16} \cdot (16 \cdot 625)$

Step 5.3.2.1.2.3

Rewrite the expression.

$x^{2} = 3 \cdot 625$

Step 5.3.2.1.3

Multiply $3$ by $625$ .

$x^{2} = 1875$

Step 5.4

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

$x = \pm \sqrt{1875}$

Step 5.5

Simplify $\pm \sqrt{1875}$ .

Tap for more steps...

Step 5.5.1

Rewrite $1875$ as $25^{2} \cdot 3$ .

Tap for more steps...

Step 5.5.1.1

Factor $625$ out of $1875$ .

$x = \pm \sqrt{625 (3)}$

Step 5.5.1.2

Rewrite $625$ as $25^{2}$ .

$x = \pm \sqrt{25^{2} \cdot 3}$

Step 5.5.2

Pull terms out from under the radical.

$x = \pm 25 \sqrt{3}$

Step 5.6

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

Tap for more steps...

Step 5.6.1

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

$x = 25 \sqrt{3}$

Step 5.6.2

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

$x = - 25 \sqrt{3}$

Step 5.6.3

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

$x = 25 \sqrt{3}, - 25 \sqrt{3}$

Step 6

Exclude the solutions that do not make $2 \log (4) - \log (3) + 2 \log (x) - 4 = 0$ true.

$x = 25 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 25 \sqrt{3}$

Decimal Form:

$x = 43.30127018 \dots$