Precalculus Examples

$\log (5) (x^{2} - 25) - (x - 5) = 2$

Step 1

Simplify $\log (5) (x^{2} - 25) - (x - 5)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\log (5) x^{2} + \log (5) \cdot - 25 - (x - 5) = 2$

Step 1.1.2

Multiply $\log (5) \cdot - 25$ .

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Step 1.1.2.1

Reorder $\log (5)$ and $- 25$ .

$\log (5) x^{2} - 25 \cdot \log (5) - (x - 5) = 2$

Step 1.1.2.2

Simplify $- 25 \cdot \log (5)$ by moving $25$ inside the logarithm.

$\log (5) x^{2} - \log (5^{25}) - (x - 5) = 2$

Step 1.1.3

Raise $5$ to the power of $25$ .

$\log (5) x^{2} - \log (298023223876953000) - (x - 5) = 2$

Step 1.1.4

Apply the distributive property.

$\log (5) x^{2} - \log (298023223876953000) - x - - 5 = 2$

Step 1.1.5

Multiply $- 1$ by $- 5$ .

$\log (5) x^{2} - \log (298023223876953000) - x + 5 = 2$

Step 1.2

Reorder factors in $\log (5) x^{2} - \log (298023223876953000) - x + 5$ .

$x^{2} \log (5) - \log (298023223876953000) - x + 5 = 2$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$x^{2} \log (5) - \log (298023223876953000) = x - 5 + 2$

Step 3

Add $- 5$ and $2$ .

$x^{2} \log (5) - \log (298023223876953000) = x - 3$

Step 4

Subtract $x$ from both sides of the equation.

$x^{2} \log (5) - \log (298023223876953000) - x = - 3$

Step 5

Add $3$ to both sides of the equation.

$x^{2} \log (5) - \log (298023223876953000) - x + 3 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = \log (5)$ , $b = - 1$ , and $c = - \log (298023223876953000) + 3$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (\log (5) \cdot (- \log (298023223876953000) + 3))}}{2 \log (5)}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot \log (5) \cdot (- \log (298023223876953000) + 3)}}{2 \log (5)}$

Step 8.2

Apply the distributive property.

$x = \frac{1 \pm \sqrt{1 - 4 \log (5) (- \log (298023223876953000)) - 4 \log (5) \cdot 3}}{2 \log (5)}$

Step 8.3

Rewrite using the commutative property of multiplication.

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 1 \log (5) \log (298023223876953000)) - 4 \log (5) \cdot 3}}{2 \log (5)}$

Step 8.4

Multiply $3$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 1 \log (5) \log (298023223876953000)) - 12 \log (5)}}{2 \log (5)}$

Step 8.5

Multiply $- 4$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 4 \log (5) \log (298023223876953000) - 12 \log (5)}}{2 \log (5)}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{1 + 4 \log (5) \log (298023223876953000) - 12 \log (5)}}{2 \log (5)}, \frac{1 - \sqrt{1 + 4 \log (5) \log (298023223876953000) - 12 \log (5)}}{2 \log (5)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{1 + 4 \log (5) \log (298023223876953000) - 12 \log (5)}}{2 \log (5)}, \frac{1 - \sqrt{1 + 4 \log (5) \log (298023223876953000) - 12 \log (5)}}{2 \log (5)}$

Decimal Form:

$x = 5.32182057 \dots, - 3.89114402 \dots$