Algebra Examples

$\log_{12} (n + 14) + \frac{1}{2} \cdot \log_{12} (4 n^{2}) = 2$

Step 1

Simplify the left side.

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Step 1.1

Combine $\frac{1}{2}$ and $\log_{12} (4 n^{2})$ .

$\log_{12} (n + 14) + \frac{\log_{12} (4 n^{2})}{2} = 2$

Step 2

To write $\log_{12} (n + 14)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{12} (n + 14) \cdot \frac{2}{2} + \frac{\log_{12} (4 n^{2})}{2} = 2$

Step 3

Simplify terms.

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Step 3.1

Combine $\log_{12} (n + 14)$ and $\frac{2}{2}$ .

$\frac{\log_{12} (n + 14) \cdot 2}{2} + \frac{\log_{12} (4 n^{2})}{2} = 2$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\log_{12} (n + 14) \cdot 2 + \log_{12} (4 n^{2})}{2} = 2$

Step 4

Move $2$ to the left of $\log_{12} (n + 14)$ .

$\frac{2 \log_{12} (n + 14) + \log_{12} (4 n^{2})}{2} = 2$

Step 5

Simplify the left side.

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Step 5.1

Simplify $\frac{2 \log_{12} (n + 14) + \log_{12} (4 n^{2})}{2}$ .

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Step 5.1.1

Simplify the numerator.

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Step 5.1.1.1

Simplify $2 \log_{12} (n + 14)$ by moving $2$ inside the logarithm.

$\frac{\log_{12} ({(n + 14)}^{2}) + \log_{12} (4 n^{2})}{2} = 2$

Step 5.1.1.2

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

$\frac{\log_{12} ({(n + 14)}^{2} (4 n^{2}))}{2} = 2$

Step 5.1.1.3

Rewrite using the commutative property of multiplication.

$\frac{\log_{12} (4 {(n + 14)}^{2} n^{2})}{2} = 2$

Step 5.1.2

Rewrite $\frac{\log_{12} (4 {(n + 14)}^{2} n^{2})}{2}$ as $\frac{1}{2} \log_{12} (4 {(n + 14)}^{2} n^{2})$ .

$\frac{1}{2} \log_{12} (4 {(n + 14)}^{2} n^{2}) = 2$

Step 5.1.3

Simplify $\frac{1}{2} \log_{12} (4 {(n + 14)}^{2} n^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{12} ({(4 {(n + 14)}^{2} n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.4.1

Apply the product rule to $4 {(n + 14)}^{2} n^{2}$ .

$\log_{12} ({(4 {(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.4.2

Apply the product rule to $4 {(n + 14)}^{2}$ .

$\log_{12} (4^{\frac{1}{2}} {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.5

Simplify the expression.

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Step 5.1.5.1

Rewrite $4$ as $2^{2}$ .

$\log_{12} ({(2^{2})}^{\frac{1}{2}} {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{12} (2^{2 (\frac{1}{2})} {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.6

Cancel the common factor of $2$ .

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Step 5.1.6.1

Cancel the common factor.

$\log_{12} (2^{2 (\frac{1}{2})} {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.6.2

Rewrite the expression.

$\log_{12} (2^{1} {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.7

Evaluate the exponent.

$\log_{12} (2 {({(n + 14)}^{2})}^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.8

Multiply the exponents in ${({(n + 14)}^{2})}^{\frac{1}{2}}$ .

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Step 5.1.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{12} (2 {(n + 14)}^{2 (\frac{1}{2})} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.8.2

Cancel the common factor of $2$ .

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Step 5.1.8.2.1

Cancel the common factor.

$\log_{12} (2 {(n + 14)}^{2 (\frac{1}{2})} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.8.2.2

Rewrite the expression.

$\log_{12} (2 {(n + 14)}^{1} {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.9

Simplify.

$\log_{12} (2 (n + 14) {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.10

Apply the distributive property.

$\log_{12} ((2 n + 2 \cdot 14) {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.11

Simplify the expression.

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Step 5.1.11.1

Multiply $2$ by $14$ .

$\log_{12} ((2 n + 28) {(n^{2})}^{\frac{1}{2}}) = 2$

Step 5.1.11.2

Multiply the exponents in ${(n^{2})}^{\frac{1}{2}}$ .

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Step 5.1.11.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{12} ((2 n + 28) n^{2 (\frac{1}{2})}) = 2$

Step 5.1.11.2.2

Cancel the common factor of $2$ .

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Step 5.1.11.2.2.1

Cancel the common factor.

$\log_{12} ((2 n + 28) n^{2 (\frac{1}{2})}) = 2$

Step 5.1.11.2.2.2

Rewrite the expression.

$\log_{12} ((2 n + 28) n^{1}) = 2$

Step 5.1.12

Simplify.

$\log_{12} ((2 n + 28) n) = 2$

Step 5.1.13

Apply the distributive property.

$\log_{12} (2 n \cdot n + 28 n) = 2$

Step 5.1.14

Multiply $n$ by $n$ by adding the exponents.

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Step 5.1.14.1

Move $n$ .

$\log_{12} (2 (n \cdot n) + 28 n) = 2$

Step 5.1.14.2

Multiply $n$ by $n$ .

$\log_{12} (2 n^{2} + 28 n) = 2$

Step 6

Rewrite $\log_{12} (2 n^{2} + 28 n) = 2$ 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 $b^{y} = x$ .

$12^{2} = 2 n^{2} + 28 n$

Step 7

Solve for $n$ .

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Step 7.1

Rewrite the equation as $2 n^{2} + 28 n = 12^{2}$ .

$2 n^{2} + 28 n = 12^{2}$

Step 7.2

Raise $12$ to the power of $2$ .

$2 n^{2} + 28 n = 144$

Step 7.3

Subtract $144$ from both sides of the equation.

$2 n^{2} + 28 n - 144 = 0$

Step 7.4

Factor the left side of the equation.

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Step 7.4.1

Factor $2$ out of $2 n^{2} + 28 n - 144$ .

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Step 7.4.1.1

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

$2 (n^{2}) + 28 n - 144 = 0$

Step 7.4.1.2

Factor $2$ out of $28 n$ .

$2 (n^{2}) + 2 (14 n) - 144 = 0$

Step 7.4.1.3

Factor $2$ out of $- 144$ .

$2 n^{2} + 2 (14 n) + 2 \cdot - 72 = 0$

Step 7.4.1.4

Factor $2$ out of $2 n^{2} + 2 (14 n)$ .

$2 (n^{2} + 14 n) + 2 \cdot - 72 = 0$

Step 7.4.1.5

Factor $2$ out of $2 (n^{2} + 14 n) + 2 \cdot - 72$ .

$2 (n^{2} + 14 n - 72) = 0$

Step 7.4.2

Factor.

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Step 7.4.2.1

Factor $n^{2} + 14 n - 72$ using the AC method.

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Step 7.4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 72$ and whose sum is $14$ .

$- 4, 18$

Step 7.4.2.1.2

Write the factored form using these integers.

$2 ((n - 4) (n + 18)) = 0$

Step 7.4.2.2

Remove unnecessary parentheses.

$2 (n - 4) (n + 18) = 0$

Step 7.5

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

$n - 4 = 0$

$n + 18 = 0$

Step 7.6

Set $n - 4$ equal to $0$ and solve for $n$ .

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Step 7.6.1

Set $n - 4$ equal to $0$ .

$n - 4 = 0$

Step 7.6.2

Add $4$ to both sides of the equation.

$n = 4$

Step 7.7

Set $n + 18$ equal to $0$ and solve for $n$ .

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Step 7.7.1

Set $n + 18$ equal to $0$ .

$n + 18 = 0$

Step 7.7.2

Subtract $18$ from both sides of the equation.

$n = - 18$

Step 7.8

The final solution is all the values that make $2 (n - 4) (n + 18) = 0$ true.

$n = 4, - 18$

Step 8

Exclude the solutions that do not make $\log_{12} (n + 14) + \frac{1}{2} \cdot \log_{12} (4 n^{2}) = 2$ true.

$n = 4$