Solve for x log of 2x+ log of x=11

$\log (2 x) + \log (x) = 11$

Step 1

Simplify the left side.

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

$\log (2 x \cdot x) = 11$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\log (2 (x \cdot x)) = 11$

Multiply $x$ by $x$ .

$\log (2 x^{2}) = 11$

Step 2

Rewrite $\log (2 x^{2}) = 11$ 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 $b^{y} = x$ .

$10^{11} = 2 x^{2}$

Step 3

Solve for $x$ .

Rewrite the equation as $2 x^{2} = 10^{11}$ .

$2 x^{2} = 10^{11}$

Divide each term in $2 x^{2} = 10^{11}$ by $2$ and simplify.

Divide each term in $2 x^{2} = 10^{11}$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{10^{11}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{10^{11}}{2}$

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

$x^{2} = \frac{10^{11}}{2}$

Simplify the right side.

Raise $10$ to the power of $11$ .

$x^{2} = \frac{100000000000}{2}$

Divide $100000000000$ by $2$ .

$x^{2} = 50000000000$

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

$x = \pm \sqrt{50000000000}$

Simplify $\pm \sqrt{50000000000}$ .

Rewrite $50000000000$ as $100000^{2} \cdot 5$ .

Factor $10000000000$ out of $50000000000$ .

$x = \pm \sqrt{10000000000 (5)}$

Rewrite $10000000000$ as $100000^{2}$ .

$x = \pm \sqrt{100000^{2} \cdot 5}$

Pull terms out from under the radical.

$x = \pm 100000 \sqrt{5}$

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.

$x = 100000 \sqrt{5}$

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

$x = - 100000 \sqrt{5}$

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

$x = 100000 \sqrt{5}, - 100000 \sqrt{5}$

Step 4

Exclude the solutions that do not make $\log (2 x) + \log (x) = 11$ true.

$x = 100000 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 100000 \sqrt{5}$

Decimal Form:

$x = 223606.79774997 \dots$

Solve for x log of 2x+ log of x=11

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