Basic Math Examples

$6 \log (n + 4) = 2 \log (8)$

Step 1

Simplify.

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Step 1.1

Simplify $6 \log (n + 4)$ by moving $6$ inside the logarithm.

$\log ({(n + 4)}^{6}) = 2 \log (8)$

Step 1.2

Simplify $2 \log (8)$ by moving $2$ inside the logarithm.

$\log ({(n + 4)}^{6}) = \log (8^{2})$

Step 2

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

${(n + 4)}^{6} = 8^{2}$

Step 3

Solve for $n$ .

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Step 3.1

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

$n + 4 = \pm \sqrt[6]{8^{2}}$

Step 3.2

Simplify $\pm \sqrt[6]{8^{2}}$ .

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Step 3.2.1

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

$n + 4 = \pm \sqrt[6]{64}$

Step 3.2.2

Rewrite $64$ as $2^{6}$ .

$n + 4 = \pm \sqrt[6]{2^{6}}$

Step 3.2.3

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

$n + 4 = \pm 2$

Step 3.3

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

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Step 3.3.1

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

$n + 4 = 2$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $4$ from both sides of the equation.

$n = 2 - 4$

Step 3.3.2.2

Subtract $4$ from $2$ .

$n = - 2$

Step 3.3.3

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

$n + 4 = - 2$

Step 3.3.4

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

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Step 3.3.4.1

Subtract $4$ from both sides of the equation.

$n = - 2 - 4$

Step 3.3.4.2

Subtract $4$ from $- 2$ .

$n = - 6$

Step 3.3.5

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

$n = - 2, - 6$

Step 4

Exclude the solutions that do not make $6 \log (n + 4) = 2 \log (8)$ true.

$n = - 2$