Solve for x 2 log base 7 of x=- log base 7 of 49

$2 \log_{7} (x) = - \log_{7} (49)$

Step 1

Simplify.

Simplify $2 \log_{7} (x)$ by moving $2$ inside the logarithm.

$\log_{7} (x^{2}) = - \log_{7} (49)$

Simplify $- \log_{7} (49)$ by moving $- 1$ inside the logarithm.

$\log_{7} (x^{2}) = \log_{7} (49^{- 1})$

Step 2

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

$x^{2} = 49^{- 1}$

Step 3

Solve for $x$ .

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

$x = \pm \sqrt{49^{- 1}}$

Simplify $\pm \sqrt{49^{- 1}}$ .

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \pm \sqrt{\frac{1}{49}}$

Rewrite $\sqrt{\frac{1}{49}}$ as $\frac{\sqrt{1}}{\sqrt{49}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{49}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{49}}$

Simplify the denominator.

Rewrite $49$ as $7^{2}$ .

$x = \pm \frac{1}{\sqrt{7^{2}}}$

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

$x = \pm \frac{1}{7}$

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.

$x = \frac{1}{7}$

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

$x = - \frac{1}{7}$

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

$x = \frac{1}{7}, - \frac{1}{7}$

Step 4

Exclude the solutions that do not make $2 \log_{7} (x) = - \log_{7} (49)$ true.

$x = \frac{1}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{7}$

Decimal Form:

$x = 0.\overline{142857}$