Solve for x log base 12 of x=1/2* log base 12 of 9+1/3* log base 12 of 27

$\log_{12} (x) = \frac{1}{2} \cdot \log_{12} (9) + \frac{1}{3} \cdot \log_{12} (27)$

Step 1

Simplify the right side.

Simplify each term.

Combine $\frac{1}{2}$ and $\log_{12} (9)$ .

$\log_{12} (x) = \frac{\log_{12} (9)}{2} + \frac{1}{3} \log_{12} (27)$

Combine $\frac{1}{3}$ and $\log_{12} (27)$ .

$\log_{12} (x) = \frac{\log_{12} (9)}{2} + \frac{\log_{12} (27)}{3}$

Step 2

Multiply each term in $\log_{12} (x) = \frac{\log_{12} (9)}{2} + \frac{\log_{12} (27)}{3}$ by $6$ to eliminate the fractions.

Multiply each term in $\log_{12} (x) = \frac{\log_{12} (9)}{2} + \frac{\log_{12} (27)}{3}$ by $6$ .

$\log_{12} (x) \cdot 6 = \frac{\log_{12} (9)}{2} \cdot 6 + \frac{\log_{12} (27)}{3} \cdot 6$

Simplify the left side.

Move $6$ to the left of $\log_{12} (x)$ .

$6 \log_{12} (x) = \frac{\log_{12} (9)}{2} \cdot 6 + \frac{\log_{12} (27)}{3} \cdot 6$

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$6 \log_{12} (x) = \frac{\log_{12} (9)}{2} \cdot (2 (3)) + \frac{\log_{12} (27)}{3} \cdot 6$

Cancel the common factor.

$6 \log_{12} (x) = \frac{\log_{12} (9)}{2} \cdot (2 \cdot 3) + \frac{\log_{12} (27)}{3} \cdot 6$

Rewrite the expression.

$6 \log_{12} (x) = \log_{12} (9) \cdot 3 + \frac{\log_{12} (27)}{3} \cdot 6$

Move $3$ to the left of $\log_{12} (9)$ .

$6 \log_{12} (x) = 3 \cdot \log_{12} (9) + \frac{\log_{12} (27)}{3} \cdot 6$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$6 \log_{12} (x) = 3 \log_{12} (9) + \frac{\log_{12} (27)}{3} \cdot (3 (2))$

Cancel the common factor.

$6 \log_{12} (x) = 3 \log_{12} (9) + \frac{\log_{12} (27)}{3} \cdot (3 \cdot 2)$

Rewrite the expression.

$6 \log_{12} (x) = 3 \log_{12} (9) + \log_{12} (27) \cdot 2$

Move $2$ to the left of $\log_{12} (27)$ .

$6 \log_{12} (x) = 3 \log_{12} (9) + 2 \log_{12} (27)$

Step 3

Simplify the left side.

Simplify $6 \log_{12} (x)$ by moving $6$ inside the logarithm.

$\log_{12} (x^{6}) = 3 \log_{12} (9) + 2 \log_{12} (27)$

Step 4

Simplify the right side.

Simplify $3 \log_{12} (9) + 2 \log_{12} (27)$ .

Simplify each term.

Simplify $3 \log_{12} (9)$ by moving $3$ inside the logarithm.

$\log_{12} (x^{6}) = \log_{12} (9^{3}) + 2 \log_{12} (27)$

Raise $9$ to the power of $3$ .

$\log_{12} (x^{6}) = \log_{12} (729) + 2 \log_{12} (27)$

Simplify $2 \log_{12} (27)$ by moving $2$ inside the logarithm.

$\log_{12} (x^{6}) = \log_{12} (729) + \log_{12} (27^{2})$

Raise $27$ to the power of $2$ .

$\log_{12} (x^{6}) = \log_{12} (729) + \log_{12} (729)$

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

$\log_{12} (x^{6}) = \log_{12} (729 \cdot 729)$

Multiply $729$ by $729$ .

$\log_{12} (x^{6}) = \log_{12} (531441)$

Step 5

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{6} = 531441$

Step 6

Solve for $x$ .

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{531441}$

Simplify $\pm \sqrt[6]{531441}$ .

Rewrite $531441$ as $9^{6}$ .

$x = \pm \sqrt[6]{9^{6}}$

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

$x = \pm 9$

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.

$x = 9$

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

$x = - 9$

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

$x = 9, - 9$

Step 7

Exclude the solutions that do not make $\log_{12} (x) = \frac{1}{2} \cdot \log_{12} (9) + \frac{1}{3} \cdot \log_{12} (27)$ true.

$x = 9$

Solve for x log base 12 of x=1/2* log base 12 of 9+1/3* log base 12 of 27

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