Solve 64^12div4^15=64^x

$${ 64 }^{ 12 } \div { 4 }^{ 15 } = { 64 }^{ x }$$

Solve for x

$$x=7$$

Steps Using Definition of Logarithm

Calculate 64 to the power of 12 and get 4722366482869645213696.

$$\frac{4722366482869645213696}{4^{15}}=64^{x}$$

Calculate 4 to the power of 15 and get 1073741824.

$$\frac{4722366482869645213696}{1073741824}=64^{x}$$

Divide 4722366482869645213696 by 1073741824 to get 4398046511104.

$$4398046511104=64^{x}$$

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

$$64^{x}=4398046511104$$

Take the logarithm of both sides of the equation.

$$\log(64^{x})=\log(4398046511104)$$

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

$$x\log(64)=\log(4398046511104)$$

Divide both sides by \log(64).

$$x=\frac{\log(4398046511104)}{\log(64)}$$

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

$$x=\log_{64}\left(4398046511104\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{3\ln(2)}+7$$

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