Solve for n (c^-9)/(c^5)=1/(c^n)

$\frac{c^{- 9}}{c^{5}} = \frac{1}{c^{n}}$

Step 1

Rewrite the equation as $\frac{1}{c^{n}} = \frac{c^{- 9}}{c^{5}}$ .

$\frac{1}{c^{n}} = \frac{c^{- 9}}{c^{5}}$

Step 2

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

$\ln (\frac{1}{c^{n}}) = \ln (\frac{c^{- 9}}{c^{5}})$

Step 3

Expand the left side.

Rewrite $\ln (\frac{1}{c^{n}})$ as $\ln (1) - \ln (c^{n})$ .

$\ln (1) - \ln (c^{n}) = \ln (\frac{c^{- 9}}{c^{5}})$

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

$\ln (1) - (n \ln (c)) = \ln (\frac{c^{- 9}}{c^{5}})$

The natural logarithm of $1$ is $0$ .

$0 - (n \ln (c)) = \ln (\frac{c^{- 9}}{c^{5}})$

Subtract $n \ln (c)$ from $0$ .

$- (n \ln (c)) = \ln (\frac{c^{- 9}}{c^{5}})$

Remove parentheses.

$- n \ln (c) = \ln (\frac{c^{- 9}}{c^{5}})$

Step 4

Simplify the right side.

Simplify $\ln (\frac{c^{- 9}}{c^{5}})$ .

Move $c^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- n \ln (c) = \ln (\frac{1}{c^{5} c^{9}})$

Multiply $c^{5}$ by $c^{9}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- n \ln (c) = \ln (\frac{1}{c^{5 + 9}})$

Add $5$ and $9$ .

$- n \ln (c) = \ln (\frac{1}{c^{14}})$

Step 5

Divide each term in $- n \ln (c) = \ln (\frac{1}{c^{14}})$ by $- 1 \ln (c)$ and simplify.

Divide each term in $- n \ln (c) = \ln (\frac{1}{c^{14}})$ by $- 1 \ln (c)$ .

$\frac{- n \ln (c)}{- 1 \ln (c)} = \frac{\ln (\frac{1}{c^{14}})}{- 1 \ln (c)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{n \ln (c)}{\ln (c)} = \frac{\ln (\frac{1}{c^{14}})}{- 1 \ln (c)}$

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

Cancel the common factor.

$\frac{n \ln (c)}{\ln (c)} = \frac{\ln (\frac{1}{c^{14}})}{- 1 \ln (c)}$

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{1}{c^{14}})}{- 1 \ln (c)}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{\ln (\frac{1}{c^{14}})}{\ln (c)}$