Finite Math Examples

$5^{x^{2} + x} = 1$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 2

Expand $\ln (5^{x^{2} + x})$ by moving $x^{2} + x$ outside the logarithm.

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

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

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

Step 4

Simplify the right side.

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Step 4.1

The natural logarithm of $1$ is $0$ .

$x^{2} \ln (5) + x \ln (5) = 0$

Step 5

Factor $x \ln (5)$ out of $x^{2} \ln (5) + x \ln (5)$ .

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Step 5.1

Factor $x \ln (5)$ out of $x^{2} \ln (5)$ .

$x \ln (5) x + x \ln (5) = 0$

Step 5.2

Factor $x \ln (5)$ out of $x \ln (5)$ .

$x \ln (5) x + x \ln (5) \cdot 1 = 0$

Step 5.3

Factor $x \ln (5)$ out of $x \ln (5) x + x \ln (5) \cdot 1$ .

$x \ln (5) (x + 1) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 1 = 0$

Step 7

Set $x$ equal to $0$ .

$x = 0$

Step 8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

The final solution is all the values that make $x \ln (5) (x + 1) = 0$ true.

$x = 0, - 1$