Solve for x 2* log of x-3 = log of 25

$2 \cdot \log (x - 3) = \log (25)$

Step 1

Simplify $2 \cdot \log (x - 3)$ by moving $2$ inside the logarithm.

$\log ({(x - 3)}^{2}) = \log (25)$

Step 2

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

${(x - 3)}^{2} = 25$

Step 3

Solve for $x$ .

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

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

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$x - 3 = \pm \sqrt{5^{2}}$

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

$x - 3 = \pm 5$

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.

$x - 3 = 5$

Move all terms not containing $x$ to the right side of the equation.

Add $3$ to both sides of the equation.

$x = 5 + 3$

Add $5$ and $3$ .

$x = 8$

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

$x - 3 = - 5$

Move all terms not containing $x$ to the right side of the equation.

Add $3$ to both sides of the equation.

$x = - 5 + 3$

Add $- 5$ and $3$ .

$x = - 2$

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

$x = 8, - 2$

Step 4

Exclude the solutions that do not make $2 \cdot \log (x - 3) = \log (25)$ true.

$x = 8$