Solve by Factoring 2 log of 4- log of 3+2 log of x-4=0

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

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

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

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

$\log (16) - \log (3) + 2 \log (x) - 4 = 0$

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

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

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

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

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

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

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

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

Add $4$ to both sides of the equation.

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

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 $b^{y} = x$ .

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

Solve for $x$ .

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

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Combine.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $16$ .

Cancel the common factor.

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

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

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

Simplify the right side.

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

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

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

Cancel the common factor of $16$ .

Factor $16$ out of $10000$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $625$ .

$x^{2} = 1875$

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

$x = \pm \sqrt{1875}$

Simplify $\pm \sqrt{1875}$ .

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

Factor $625$ out of $1875$ .

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

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

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

Pull terms out from under the radical.

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

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

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

$x = 25 \sqrt{3}$

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

$x = - 25 \sqrt{3}$

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 43.30127018 \dots, - 43.30127018 \dots$

Solve by Factoring 2 log of 4- log of 3+2 log of x-4=0

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