Calculus Examples

$10^{x^{2} - 1} = 1$

Step 1

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{x^{2} - 1}) = \log (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\log (10^{x^{2} - 1})$ by moving $x^{2} - 1$ outside the logarithm.

$(x^{2} - 1) \log (10) = \log (1)$

Step 2.2

Logarithm base $10$ of $10$ is $1$ .

$(x^{2} - 1) \cdot 1 = \log (1)$

Step 2.3

Multiply $x^{2} - 1$ by $1$ .

$x^{2} - 1 = \log (1)$

Step 3

Simplify the right side.

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Step 3.1

Logarithm base $10$ of $1$ is $0$ .

$x^{2} - 1 = 0$

Step 4

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 5

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

$x = \pm \sqrt{1}$

Step 6

Any root of $1$ is $1$ .

$x = \pm 1$

Step 7

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

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Step 7.1

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

$x = 1$

Step 7.2

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

$x = - 1$

Step 7.3

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

$x = 1, - 1$