Algebra Examples

$\log_{8} (p^{2} + 15) = 2$

Step 1

Rewrite $\log_{8} (p^{2} + 15) = 2$ 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 $b^{y} = x$ .

$8^{2} = p^{2} + 15$

Step 2

Solve for $p$ .

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Step 2.1

Rewrite the equation as $p^{2} + 15 = 8^{2}$ .

$p^{2} + 15 = 8^{2}$

Step 2.2

Raise $8$ to the power of $2$ .

$p^{2} + 15 = 64$

Step 2.3

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

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Step 2.3.1

Subtract $15$ from both sides of the equation.

$p^{2} = 64 - 15$

Step 2.3.2

Subtract $15$ from $64$ .

$p^{2} = 49$

Step 2.4

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

$p = \pm \sqrt{49}$

Step 2.5

Simplify $\pm \sqrt{49}$ .

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Step 2.5.1

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

$p = \pm \sqrt{7^{2}}$

Step 2.5.2

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

$p = \pm 7$

Step 2.6

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

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Step 2.6.1

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

$p = 7$

Step 2.6.2

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

$p = - 7$

Step 2.6.3

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

$p = 7, - 7$