Solve 8^n-3n-9=9^3

$$8 ^ { n - 3 n - 9 } = 9 ^ { 3 }$$

Solve for n

$$n=-\log_{2}\left(3\right)-\frac{9}{2}\approx -6.084962501$$

Steps Using Definition of Logarithm

Combine n and -3n to get -2n.

$$8^{-2n-9}=9^{3}$$

Calculate 9 to the power of 3 and get 729.

$$8^{-2n-9}=729$$

Take the logarithm of both sides of the equation.

$$\log(8^{-2n-9})=\log(729)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(-2n-9\right)\log(8)=\log(729)$$

Divide both sides by \log(8).

$$-2n-9=\frac{\log(729)}{\log(8)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-2n-9=\log_{8}\left(729\right)$$

Add 9 to both sides of the equation.

$$-2n=2\log_{2}\left(3\right)-\left(-9\right)$$

Divide both sides by -2.

$$n=\frac{2\log_{2}\left(3\right)+9}{-2}$$