Algebra Examples

$\log_{5} ({(2 x - 3)}^{2}) = 6$

Step 1

Write in exponential form.

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Step 1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 5$ , $x = {(2 x - 3)}^{2}$ , and $y = 6$ .

$b = 5$

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

$y = 6$

Step 1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$5^{6} = {(2 x - 3)}^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as ${(2 x - 3)}^{2} = 5^{6}$ .

${(2 x - 3)}^{2} = 5^{6}$

Step 2.2

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

$2 x - 3 = \pm \sqrt{5^{6}}$

Step 2.3

Simplify $\pm \sqrt{5^{6}}$ .

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Step 2.3.1

Raise $5$ to the power of $6$ .

$2 x - 3 = \pm \sqrt{15625}$

Step 2.3.2

Rewrite $15625$ as $125^{2}$ .

$2 x - 3 = \pm \sqrt{125^{2}}$

Step 2.3.3

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

$2 x - 3 = \pm 125$

Step 2.4

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

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Step 2.4.1

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

$2 x - 3 = 125$

Step 2.4.2

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

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Step 2.4.2.1

Add $3$ to both sides of the equation.

$2 x = 125 + 3$

Step 2.4.2.2

Add $125$ and $3$ .

$2 x = 128$

Step 2.4.3

Divide each term in $2 x = 128$ by $2$ and simplify.

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Step 2.4.3.1

Divide each term in $2 x = 128$ by $2$ .

$\frac{2 x}{2} = \frac{128}{2}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{128}{2}$

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{128}{2}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Divide $128$ by $2$ .

$x = 64$

Step 2.4.4

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

$2 x - 3 = - 125$

Step 2.4.5

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

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Step 2.4.5.1

Add $3$ to both sides of the equation.

$2 x = - 125 + 3$

Step 2.4.5.2

Add $- 125$ and $3$ .

$2 x = - 122$

Step 2.4.6

Divide each term in $2 x = - 122$ by $2$ and simplify.

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Step 2.4.6.1

Divide each term in $2 x = - 122$ by $2$ .

$\frac{2 x}{2} = \frac{- 122}{2}$

Step 2.4.6.2

Simplify the left side.

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Step 2.4.6.2.1

Cancel the common factor of $2$ .

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Step 2.4.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 122}{2}$

Step 2.4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 122}{2}$

Step 2.4.6.3

Simplify the right side.

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Step 2.4.6.3.1

Divide $- 122$ by $2$ .

$x = - 61$

Step 2.4.7

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

$x = 64, - 61$