Precalculus Examples

$2 \log_{5} (x + 4) = \log_{5} (5 + 1)$

Step 1

Simplify $2 \log_{5} (x + 4)$ by moving $2$ inside the logarithm.

$\log_{5} ({(x + 4)}^{2}) = \log_{5} (5 + 1)$

Step 2

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

${(x + 4)}^{2} = 5 + 1$

Step 3

Solve for $x$ .

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Step 3.1

Add $5$ and $1$ .

${(x + 4)}^{2} = 6$

Step 3.2

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

$x + 4 = \pm \sqrt{6}$

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.

$x + 4 = \sqrt{6}$

Step 3.3.2

Subtract $4$ from both sides of the equation.

$x = \sqrt{6} - 4$

Step 3.3.3

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

$x + 4 = - \sqrt{6}$

Step 3.3.4

Subtract $4$ from both sides of the equation.

$x = - \sqrt{6} - 4$

Step 3.3.5

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

$x = \sqrt{6} - 4, - \sqrt{6} - 4$

Step 4

Exclude the solutions that do not make $2 \log_{5} (x + 4) = \log_{5} (5 + 1)$ true.

$x = \sqrt{6} - 4$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{6} - 4$

Decimal Form:

$x = - 1.55051025 \dots$