Solve for n f(x)=7x^(-3n)+x^2

$f (x) = 7 x^{- 3 n} + x^{2}$

Step 1

Rewrite the equation as $7 x^{- 3 n} + x^{2} = f (x)$ .

$7 x^{- 3 n} + x^{2} = f (x)$

Step 2

Rewrite $x^{- 3 n}$ as exponentiation.

$7 {(x^{n})}^{- 3} + x^{2} = f (x)$

Step 3

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

$7 x^{- 3 n} = f (x) - x^{2}$

Step 4

Divide each term in $7 x^{- 3 n} = f (x) - x^{2}$ by $7$ and simplify.

Divide each term in $7 x^{- 3 n} = f (x) - x^{2}$ by $7$ .

$\frac{7 x^{- 3 n}}{7} = \frac{f (x)}{7} + \frac{- x^{2}}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{- 3 n}}{7} = \frac{f (x)}{7} + \frac{- x^{2}}{7}$

Divide $x^{- 3 n}$ by $1$ .

$x^{- 3 n} = \frac{f (x)}{7} + \frac{- x^{2}}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{- 3 n} = \frac{f (x)}{7} - \frac{x^{2}}{7}$

Step 5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (x^{- 3 n}) = \ln (\frac{f (x)}{7} - \frac{x^{2}}{7})$

Step 6

Expand $\ln (x^{- 3 n})$ by moving $- 3 n$ outside the logarithm.

$- 3 n \ln (x) = \ln (\frac{f (x)}{7} - \frac{x^{2}}{7})$

Step 7

Divide each term in $- 3 n \ln (x) = \ln (\frac{f (x)}{7} - \frac{x^{2}}{7})$ by $- 3 \ln (x)$ and simplify.

Divide each term in $- 3 n \ln (x) = \ln (\frac{f (x)}{7} - \frac{x^{2}}{7})$ by $- 3 \ln (x)$ .

$\frac{- 3 n \ln (x)}{- 3 \ln (x)} = \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{- 3 \ln (x)}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 n \ln (x)}{- 3 \ln (x)} = \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{- 3 \ln (x)}$

Rewrite the expression.

$\frac{n \ln (x)}{\ln (x)} = \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{- 3 \ln (x)}$

Cancel the common factor of $\ln (x)$ .

Cancel the common factor.

$\frac{n \ln (x)}{\ln (x)} = \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{- 3 \ln (x)}$

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{- 3 \ln (x)}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{\ln (\frac{f (x)}{7} - \frac{x^{2}}{7})}{3 \ln (x)}$

Solve for n f(x)=7x^(-3n)+x^2

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