Solve 8^4+4^3=4^m

$$8 ^ { 4 } + 4 ^ { 3 } = 4 ^ { m }$$

Solve for m

$$m = \frac{\log_{2} {(65)} + 6}{2} \approx 6.011183907$$

Steps Using Definition of Logarithm

Calculate 8 to the power of 4 and get 4096.

$$4096+4^{3}=4^{m}$$

Calculate 4 to the power of 3 and get 64.

$$4096+64=4^{m}$$

Add 4096 and 64 to get 4160.

$$4160=4^{m}$$

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

$$4^{m}=4160$$

Take the logarithm of both sides of the equation.

$$\log(4^{m})=\log(4160)$$

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

$$m\log(4)=\log(4160)$$

Divide both sides by \log(4).

$$m=\frac{\log(4160)}{\log(4)}$$

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

$$m=\log_{4}\left(4160\right)$$

Solve for m (complex solution)

$$m=\frac{\pi n_{1}i}{\ln(2)}+\frac{\log_{2}\left(4160\right)}{2}$$

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