Solve for a 2* natural log of a+3=1/4* natural log of 16+ natural log of a+7

$2 \cdot \ln (a + 3) = \frac{1}{4} \cdot \ln (16) + \ln (a + 7)$

Simplify the expressions in the equation.

Simplify the left side.

$2 \ln (a + 3) = \frac{1}{4} \ln (16) + \ln (a + 7)$

Simplify the right side.

Combine $\frac{1}{4}$ and $\ln (16)$ .

$2 \ln (a + 3) = \frac{\ln (16)}{4} + \ln (a + 7)$

Simplify the left side.

Simplify $2 \ln (a + 3)$ by moving $2$ inside the logarithm.

$\ln ({(a + 3)}^{2}) = \frac{\ln (16)}{4} + \ln (a + 7)$

Simplify the right side.

Simplify $\frac{\ln (16)}{4} + \ln (a + 7)$ .

Simplify each term.

Rewrite $\ln (16)$ as $\ln (2^{4})$ .

$\ln ({(a + 3)}^{2}) = \frac{\ln (2^{4})}{4} + \ln (a + 7)$

Expand $\ln (2^{4})$ by moving $4$ outside the logarithm.

$\ln ({(a + 3)}^{2}) = \frac{4 \ln (2)}{4} + \ln (a + 7)$

Cancel the common factor of $4$ .

Cancel the common factor.

$\ln ({(a + 3)}^{2}) = \frac{4 \ln (2)}{4} + \ln (a + 7)$

Divide $\ln (2)$ by $1$ .

$\ln ({(a + 3)}^{2}) = \ln (2) + \ln (a + 7)$

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

$\ln ({(a + 3)}^{2}) = \ln (2 (a + 7))$

Apply the distributive property.

$\ln ({(a + 3)}^{2}) = \ln (2 a + 2 \cdot 7)$

Multiply $2$ by $7$ .

$\ln ({(a + 3)}^{2}) = \ln (2 a + 14)$

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

${(a + 3)}^{2} = 2 a + 14$

Solve for $a$ .

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

Subtract $2 a$ from both sides of the equation.

${(a + 3)}^{2} - 2 a = 14$

Simplify each term.

Rewrite ${(a + 3)}^{2}$ as $(a + 3) (a + 3)$ .

$(a + 3) (a + 3) - 2 a = 14$

Expand $(a + 3) (a + 3)$ using the FOIL Method.

Apply the distributive property.

$a (a + 3) + 3 (a + 3) - 2 a = 14$

Apply the distributive property.

$a \cdot a + a \cdot 3 + 3 (a + 3) - 2 a = 14$

Apply the distributive property.

$a \cdot a + a \cdot 3 + 3 a + 3 \cdot 3 - 2 a = 14$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

$a^{2} + a \cdot 3 + 3 a + 3 \cdot 3 - 2 a = 14$

Move $3$ to the left of $a$ .

$a^{2} + 3 \cdot a + 3 a + 3 \cdot 3 - 2 a = 14$

Multiply $3$ by $3$ .

$a^{2} + 3 a + 3 a + 9 - 2 a = 14$

Add $3 a$ and $3 a$ .

$a^{2} + 6 a + 9 - 2 a = 14$

Subtract $2 a$ from $6 a$ .

$a^{2} + 4 a + 9 = 14$

Subtract $14$ from both sides of the equation.

$a^{2} + 4 a + 9 - 14 = 0$

Subtract $14$ from $9$ .

$a^{2} + 4 a - 5 = 0$

Factor $a^{2} + 4 a - 5$ using the AC method.

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Write the factored form using these integers.

$(a - 1) (a + 5) = 0$

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

$a - 1 = 0$

$a + 5 = 0$

Set $a - 1$ equal to $0$ and solve for $a$ .

Set $a - 1$ equal to $0$ .

$a - 1 = 0$

Add $1$ to both sides of the equation.

$a = 1$

Set $a + 5$ equal to $0$ and solve for $a$ .

Set $a + 5$ equal to $0$ .

$a + 5 = 0$

Subtract $5$ from both sides of the equation.

$a = - 5$

The final solution is all the values that make $(a - 1) (a + 5) = 0$ true.

$a = 1, - 5$

Exclude the solutions that do not make $2 \cdot \ln (a + 3) = \frac{1}{4} \cdot \ln (16) + \ln (a + 7)$ true.

$a = 1$

Solve for a 2* natural log of a+3=1/4* natural log of 16+ natural log of a+7

Related Questions

Support LCMGCF.com by disabling your adblocker.