Finite Math Examples

$\log_{x \cdot x + 66} (7) = 1$

Step 1

Rewrite $\log_{x \cdot x + 66} (7) = 1$ 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 $b^{y} = x$ .

${(x \cdot x + 66)}^{1} = 7$

Step 2

Solve for $x$ .

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Step 2.1

Simplify ${(x \cdot x + 66)}^{1}$ .

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Step 2.1.1

Multiply $x$ by $x$ .

${(x^{2} + 66)}^{1} = 7$

Step 2.1.2

Simplify.

$x^{2} + 66 = 7$

Step 2.2

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

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Step 2.2.1

Subtract $66$ from both sides of the equation.

$x^{2} = 7 - 66$

Step 2.2.2

Subtract $66$ from $7$ .

$x^{2} = - 59$

Step 2.3

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

$x = \pm \sqrt{- 59}$

Step 2.4

Simplify $\pm \sqrt{- 59}$ .

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Step 2.4.1

Rewrite $- 59$ as $- 1 (59)$ .

$x = \pm \sqrt{- 1 (59)}$

Step 2.4.2

Rewrite $\sqrt{- 1 (59)}$ as $\sqrt{- 1} \cdot \sqrt{59}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{59}$

Step 2.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{59}$

Step 2.5

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

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Step 2.5.1

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

$x = i \sqrt{59}$

Step 2.5.2

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

$x = - i \sqrt{59}$

Step 2.5.3

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

$x = i \sqrt{59}, - i \sqrt{59}$