Algebra Examples

$\ln (x^{2} - 25) = 0$

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x^{2} - 25)} = e^{0}$

Step 2

Rewrite $\ln (x^{2} - 25) = 0$ 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 $b^{y} = x$ .

$e^{0} = x^{2} - 25$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - 25 = e^{0}$ .

$x^{2} - 25 = e^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

$x^{2} - 25 = 1$

Step 3.3

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

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Step 3.3.1

Add $25$ to both sides of the equation.

$x^{2} = 1 + 25$

Step 3.3.2

Add $1$ and $25$ .

$x^{2} = 26$

Step 3.4

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

$x = \pm \sqrt{26}$

Step 3.5

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

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Step 3.5.1

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

$x = \sqrt{26}$

Step 3.5.2

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

$x = - \sqrt{26}$

Step 3.5.3

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

$x = \sqrt{26}, - \sqrt{26}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{26}, - \sqrt{26}$

Decimal Form:

$x = 5.09901951 \dots, - 5.09901951 \dots$