Precalculus Examples

$\log_{15} (- 10 n - 3) = \log_{15} (n^{2} - 3)$

Step 1

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

$- 10 n - 3 = n^{2} - 3$

Step 2

Solve for $n$ .

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Step 2.1

Subtract $n^{2}$ from both sides of the equation.

$- 10 n - 3 - n^{2} = - 3$

Step 2.2

Add $3$ to both sides of the equation.

$- 10 n - 3 - n^{2} + 3 = 0$

Step 2.3

Combine the opposite terms in $- 10 n - 3 - n^{2} + 3$ .

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Step 2.3.1

Add $- 3$ and $3$ .

$- 10 n - n^{2} + 0 = 0$

Step 2.3.2

Add $- 10 n - n^{2}$ and $0$ .

$- 10 n - n^{2} = 0$

Step 2.4

Factor $- n$ out of $- 10 n - n^{2}$ .

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Step 2.4.1

Reorder $- 10 n$ and $- n^{2}$ .

$- n^{2} - 10 n = 0$

Step 2.4.2

Factor $- n$ out of $- n^{2}$ .

$- n \cdot n - 10 n = 0$

Step 2.4.3

Factor $- n$ out of $- 10 n$ .

$- n \cdot n - n \cdot 10 = 0$

Step 2.4.4

Factor $- n$ out of $- n (n) - n (10)$ .

$- n (n + 10) = 0$

Step 2.5

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

$n = 0$

$n + 10 = 0$

Step 2.6

Set $n$ equal to $0$ .

$n = 0$

Step 2.7

Set $n + 10$ equal to $0$ and solve for $n$ .

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Step 2.7.1

Set $n + 10$ equal to $0$ .

$n + 10 = 0$

Step 2.7.2

Subtract $10$ from both sides of the equation.

$n = - 10$

Step 2.8

The final solution is all the values that make $- n (n + 10) = 0$ true.

$n = 0, - 10$

Step 3

Exclude the solutions that do not make $\log_{15} (- 10 n - 3) = \log_{15} (n^{2} - 3)$ true.

$n = - 10$