Solve quad4^-3*4^4=4^n

$$\quad 4 ^ { - 3 } \times 4 ^ { 4 } = 4 ^ { n }$$

Solve for n

$$n=1$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add -3 and 4 to get 1.

$$4^{1}=4^{n}$$

Calculate 4 to the power of 1 and get 4.

$$4=4^{n}$$

Swap sides so that all variable terms are on the left hand side.

$$4^{n}=4$$

Take the logarithm of both sides of the equation.

$$\log(4^{n})=\log(4)$$

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

$$n\log(4)=\log(4)$$

Divide both sides by \log(4).

$$n=\frac{\log(4)}{\log(4)}$$

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

$$n=\log_{4}\left(4\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{\ln(2)}+1$$

$$n_{1}\in \mathrm{Z}$$