Solve 4^3*4^5=4^x/4^3

$$4 ^ { 3 } \times 4 ^ { 5 } = \frac { 4 ^ { x } } { 4 ^ { 3 } }$$

Solve for x

$$x=11$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$4^{8}=\frac{4^{x}}{4^{3}}$$

Calculate 4 to the power of 8 and get 65536.

$$65536=\frac{4^{x}}{4^{3}}$$

Calculate 4 to the power of 3 and get 64.

$$65536=\frac{4^{x}}{64}$$

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

$$\frac{4^{x}}{64}=65536$$

Multiply both sides by 64.

$$4^{x}=65536\times 64$$

Multiply 65536 and 64 to get 4194304.

$$4^{x}=4194304$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(4194304)$$

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

$$x\log(4)=\log(4194304)$$

Divide both sides by \log(4).

$$x=\frac{\log(4194304)}{\log(4)}$$

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

$$x=\log_{4}\left(4194304\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(2)}+11$$

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