Basic Math Examples

$\log (c + 1) + \log (c - 1) = 3$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((c + 1) (c - 1)) = 3$

Step 1.2

Expand $(c + 1) (c - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log (c (c - 1) + 1 (c - 1)) = 3$

Step 1.2.2

Apply the distributive property.

$\log (c \cdot c + c \cdot - 1 + 1 (c - 1)) = 3$

Step 1.2.3

Apply the distributive property.

$\log (c \cdot c + c \cdot - 1 + 1 c + 1 \cdot - 1) = 3$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $c$ by $c$ .

$\log (c^{2} + c \cdot - 1 + 1 c + 1 \cdot - 1) = 3$

Step 1.3.1.2

Move $- 1$ to the left of $c$ .

$\log (c^{2} - 1 \cdot c + 1 c + 1 \cdot - 1) = 3$

Step 1.3.1.3

Rewrite $- 1 c$ as $- c$ .

$\log (c^{2} - c + 1 c + 1 \cdot - 1) = 3$

Step 1.3.1.4

Multiply $c$ by $1$ .

$\log (c^{2} - c + c + 1 \cdot - 1) = 3$

Step 1.3.1.5

Multiply $- 1$ by $1$ .

$\log (c^{2} - c + c - 1) = 3$

Step 1.3.2

Add $- c$ and $c$ .

$\log (c^{2} + 0 - 1) = 3$

Step 1.3.3

Add $c^{2}$ and $0$ .

$\log (c^{2} - 1) = 3$

Step 2

Rewrite $\log (c^{2} - 1) = 3$ 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 $b^{y} = x$ .

$10^{3} = c^{2} - 1$

Step 3

Solve for $c$ .

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Step 3.1

Rewrite the equation as $c^{2} - 1 = 10^{3}$ .

$c^{2} - 1 = 10^{3}$

Step 3.2

Raise $10$ to the power of $3$ .

$c^{2} - 1 = 1000$

Step 3.3

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

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Step 3.3.1

Add $1$ to both sides of the equation.

$c^{2} = 1000 + 1$

Step 3.3.2

Add $1000$ and $1$ .

$c^{2} = 1001$

Step 3.4

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

$c = \pm \sqrt{1001}$

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.

$c = \sqrt{1001}$

Step 3.5.2

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

$c = - \sqrt{1001}$

Step 3.5.3

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

$c = \sqrt{1001}, - \sqrt{1001}$

Step 4

Exclude the solutions that do not make $\log (c + 1) + \log (c - 1) = 3$ true.

$c = \sqrt{1001}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$c = \sqrt{1001}$

Decimal Form:

$c = 31.63858403 \dots$