Evaluate Using the Given Value 1 , 296^(2n)=216^(n+10)

$1$ , $296^{2 n} = 216^{n + 10}$

Step 1

Take the log of both sides of the equation.

$\ln (296^{2 n}) = \ln (216^{n + 10})$

Step 2

Expand $\ln (296^{2 n})$ by moving $2 n$ outside the logarithm.

$2 n \ln (296) = \ln (216^{n + 10})$

Step 3

Expand $\ln (216^{n + 10})$ by moving $n + 10$ outside the logarithm.

$2 n \ln (296) = (n + 10) \ln (216)$

Step 4

Solve the equation for $n$ .

Apply the distributive property.

$2 n \ln (296) = n \ln (216) + 10 \ln (216)$

Subtract $n \ln (216)$ from both sides of the equation.

$2 n \ln (296) - n \ln (216) = 10 \ln (216)$

Factor $n$ out of $2 n \ln (296) - n \ln (216)$ .

Factor $n$ out of $2 n \ln (296)$ .

$n (2 \ln (296)) - n \ln (216) = 10 \ln (216)$

Factor $n$ out of $- n \ln (216)$ .

$n (2 \ln (296)) + n (- 1 \ln (216)) = 10 \ln (216)$

Factor $n$ out of $n (2 \ln (296)) + n (- 1 \ln (216))$ .

$n (2 \ln (296) - 1 \ln (216)) = 10 \ln (216)$

Rewrite $- 1 \ln (216)$ as $- \ln (216)$ .

$n (2 \ln (296) - \ln (216)) = 10 \ln (216)$

Divide each term in $n (2 \ln (296) - \ln (216)) = 10 \ln (216)$ by $2 \ln (296) - \ln (216)$ and simplify.

Divide each term in $n (2 \ln (296) - \ln (216)) = 10 \ln (216)$ by $2 \ln (296) - \ln (216)$ .

$\frac{n (2 \ln (296) - \ln (216))}{2 \ln (296) - \ln (216)} = \frac{10 \ln (216)}{2 \ln (296) - \ln (216)}$

Simplify the left side.

Cancel the common factor of $2 \ln (296) - \ln (216)$ .

Cancel the common factor.

$\frac{n (2 \ln (296) - \ln (216))}{2 \ln (296) - \ln (216)} = \frac{10 \ln (216)}{2 \ln (296) - \ln (216)}$

Divide $n$ by $1$ .

$n = \frac{10 \ln (216)}{2 \ln (296) - \ln (216)}$

Step 5

Replace the variable $n$ with $\frac{10 \ln (216)}{2 \ln (296) - \ln (216)}$ in the expression.

$1$